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Abstract

Singularity is one of the important features in invariant structures in several physical phenomena reflected often in the associated invariant differential equations. The classification problem for singular differential invariants in (1+3)-dimensional space associated with Lie algebras of dimension 4 is investigated. The formulation of singular invariants for a Lie algebra of dimension $4$ possessed by the underlying system of three second-order ordinary differential equations is studied in detail and the corresponding canonical forms for these systems are deduced. Furthermore, the categorization of singular invariants on the basis of conditional singularity, weak uncoupling, weak linearization, partial uncoupling and partial linearization are described for the underlying canonical forms. In addition, those cases of classified canonical forms are also mentioned which do not lead to singular invariant systems of three second-order ODEs for a Lie algebra of dimension 4. The integrability aspect of these classified singular-invariant systems in (1+3)-dimensional space is discussed in a detailed manner for a Lie algebra of dimension 4. Finally, two physical systems from mechanics are presented to illustrate the utilization of the physical aspect of these singular invariants.

Keywords

Systems of second-order ODEs singular invariant equations conditional singularity integrability Lie algebra

Introduction

Lie group theory is an interesting area of research in which differential equations (DEs) are analysed on the basis of their corresponding invariant structure. This theory was initiated towards the end of the 19th century by the great Norwegian mathematician Sophus Lie (1842–1899). The concept of differential invariants appeared in Lie,^1,2 further considered by Tressé and many researchers. Consequently, the theory of differential invariants became a prominent part of Lie theory. Symmetries of the underlying DEs are obtained in terms of their generators and then the corresponding Lie algebra is obtained. The invariants associated with the admitted Lie algebras are deduced and they are utilized for the analysis of the underlying DEs, inter alia, integration and classification.^{3–11,13,14,12,15,16} Most of the focus in the literature, is related to regular differential invariants of the corresponding Lie algebras of the DEs under study. Singularity of the differential invariant structure associated with the Lie algebra of the underlying DEs is rather limited and often overlooked. This demands a deeper insight on the problem of singularity in the differential invariant structure of the Lie algebras under consideration.

In 1999, the notion and concept of singular invariants has been described in terms of solution manifold in the work of Ibragimov.¹⁷ In the case of a system of ordinary DEs (ODEs), the classical direct approach of successive reduction of order, once the symmetry algebra is known, is not applicable.^3,18 Thus, many researchers decided to consider the classification of systems of ODEs via their symmetries. In this regard, Popovych et al.¹⁹ studied the realizations of finite-dimensional Lie algebras and presented an updated classification in their work. By utilising¹⁹ and employing differential invariants, Gaponova and Nesterenko⁷ classified a system of two second-order ODEs admitting Lie algebras of dimensions not greater than four in their work. Moreover, by utilizing the concept of singular invariants given in the work of Ibragimov,¹⁷ they also investigated the singular invariants for these systems of ODEs. However, the integration and other diverse features of several analysis of these underlying classified invariant systems have not been discussed in the work of Gaponova and Nesterenko.⁷ Ayub and colleagues investigated the same problem. They not only developed a new type of invariant construction but classified a system of two second-order ODEs possessing Lie algebras of dimensions 3 and 4 via regular invariants in their work.^3,4 In addition, they also presented an integration strategy on the basis of their classification. By means of the new invariant scheme of construction, they classified singular invariant systems of two second-order ODEs admitting Lie algebras of dimension 4 in Ayub et al.²⁰ Moreover, this invariant classification leads to the promulgation of integrability procedure as well as various other properties of classified invariant systems viz., conditional singularity, weak uncoupling, weak linearization, and partial uncoupling, partial linearization.

Systems of three second-order ODEs occur in many physical applications, viz. mechanics of particles, fluid mechanics, small oscillations, neural oscillations (brain wave), charged particles possessing magnetic field effects etc. Thus, the area of research related to a system of three second-order ODEs has an indispensable role in the applied sciences. Limited investigations have been performed on a system of three second-order ODEs, see e.g.^11,13,14,12 Recently, by using the same approach as developed in,³ Zahida et al. ¹² constructed a basis of second-order differential invariants in (1+3)-dimensional space admitting Lie algebras of dimension three. They also formulated systems of three second-order ODEs possessing three-dimensional Lie algebras for these underlying invariants. However, there still remain various unexplored features for a system of three second-order ODEs with respect to their symmetry algebra analysis.

Here our main interest is in singularity in differential invariant structures associated with a system of three second-order ODEs having Lie algebras of dimension $4$ . The invariant approach given in literary works^3,4,20,12 is used. Conditional singularity in the differential invariant structure is also described. An integration algorithm based on singularity for these classified invariant systems possessing a Lie algebra of dimension $4$ is presented. We finally, provide applications from the mechanics of particles to illustrate the results of this classification work.

Notation

The subsequent notations are invoked for the underlying Lie algebra with allied realizations as well as corresponding invariant structure. In the sequel, $A$ is utilized as a place holder for the Lie algebra $A_{i, j}^{a, b, n}$ which signifies the $j$ th algebra of dimension $i$ , where the superscripts $a, b$ specify parameters on which the Lie algebra depends; the column $N$ in Table 1 given at the end indicates the algebra realizations; the realization is referred to by a superscript $n$ . As usual one has $\partial_{t} = \frac{\partial}{\partial t}, \partial_{x} = \frac{\partial}{\partial x}, \partial_{y} = \frac{\partial}{\partial y}, \partial_{z} = \frac{\partial}{\partial z}$ and $X_{i}$ represents the elements of a basis for the underlying Lie algebra of vector fields; here $i$ is less than or equal to the dimension of the considered Lie algebra.

Table 1.

Singular differential invariants and equations

Algebra	$N$	Invariants
$A_{4, 1}$	2	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{1}, m_{2} = I_{2}, m_{3} = I_{3}, m_{4} = \frac{d I_{2}}{d I_{1}}, m_{5} = \frac{d I_{3}}{d I_{1}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ , provided that $f_{I_{1} I_{1}} + 2 \frac{d I_{2}}{d I_{1}} f_{I_{1} I_{2}} + {(\frac{d I_{2}}{d I_{1}})}^{2} f_{I_{2} I_{2}} = 0$ $\ddot{x} = k (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = h (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 1}$	3	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{2}, m_{3} = I_{3}, m_{4} = \frac{d I_{2}}{d I_{1}}, m_{5} = \frac{d I_{3}}{d I_{1}}, m_{6} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{7} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 2}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = \frac{I_{2}}{I_{1}}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = I_{1} J, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{1}{t} f (\frac{I_{2}}{I_{1}}, I_{3}, \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = \frac{1}{t^{2}} h (\frac{I_{2}}{I_{1}}, I_{3}, \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 3}$	1	$I_{1} = t,, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \ddot{y}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : J = \ddot{x} = 0$ $m_{1} = I_{2}, m_{2} = I_{3}, m_{3} = I_{1} \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = {I_{1}}^{2} \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = 0, \ddot{y} = \frac{{\dot{x}}^{2}}{x^{2}} g (I_{2}, I_{3}, I_{1} \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{z} = \frac{{\dot{x}}^{2}}{x^{2}} h (I_{2}, I_{3}, I_{1} \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 4}$ $μ^{'} (t) = 0$	2	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = 0$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = J, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = f (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = h (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 5}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = 0$ $m_{1} = I_{2} e^{I_{1}}, m_{2} = I_{3}, m_{3} = e^{I_{1}} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = J e^{I_{1}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = e^{- t} f (I_{2} e^{I_{1}}, I_{3}, e^{I_{1}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = h (I_{2} e^{I_{1}}, I_{3}, e^{I_{1}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 5}$	6	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 6}^{1}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{J}{I_{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = y f (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = h (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$ψ^{'} (x) = 0$	4	$I_{1} = x, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \frac{\dot{y}}{\dot{x}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{z}}{\dot{x}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{x}}^{2}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{z}}{{\dot{x}}^{2}} with condition : J = \ddot{x} = 0$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = 0, \ddot{y} = {\dot{x}}^{2} g (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{z} = {\dot{x}}^{2} h (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 6}^{a}$ $a \neq 0, 1$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{2} {I_{1}}^{\frac{- 1}{1 - a}}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}} {I_{1}}^{\frac{- a}{1 - a}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = J {I_{1}}^{\frac{1 - 2 a}{1 - a}}, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = t^{\frac{2 a - 1}{1 - a}} f (I_{2} {I_{1}}^{\frac{- 1}{1 - a}}, I_{3}, \frac{d I_{2}}{d I_{1}} {I_{1}}^{\frac{- a}{1 - a}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = h (I_{2} {I_{1}}^{\frac{- 1}{1 - a}}, I_{3}, \frac{d I_{2}}{d I_{1}} {I_{1}}^{\frac{- a}{1 - a}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 6}^{a}$ $a \neq 0, 1$	7	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{1}{I_{2}} \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + x {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 7}^{a}$	4	$I_{1} = z, I_{2} = y, I_{3} = (1 + x^{2}) \frac{\dot{z}}{\dot{x}}, \frac{d I_{2}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, J = - \frac{\ddot{x} e^{a \arctan x}}{{\dot{x}}^{\frac{3}{2}} {\dot{y}}^{\frac{3}{2}}}, \frac{d I_{3}}{d I_{1}} = 2 x + (1 + x^{2}) (\frac{\ddot{z}}{\dot{x} \dot{z}} - \frac{\ddot{x}}{{\dot{x}}^{2}}), \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}} with condition : \frac{I_{3}}{\frac{d I_{2}}{d I_{1}}} \frac{d^{2} I_{2}}{d {I_{1}}^{2}} + \frac{d I_{3}}{d I_{1}} - 2 a + I_{2} \frac{(1 + a^{2})}{I_{3} \frac{d I_{2}}{d I_{1}}} = (\frac{\ddot{y}}{\dot{x} \dot{z}} - \frac{\ddot{x}}{{\dot{x}}^{2}}) + \frac{2 (x - a)}{(1 + x^{2})} + \frac{y \dot{x} (1 + a^{2})}{\dot{y} (1 + x^{2})^{2}} = 0$ $m_{1} = I_{1}, m_{2} = I_{2}, m_{3} = I_{3}, m_{4} = \frac{d I_{2}}{d I_{1}}, m_{5} = \frac{d I_{3}}{d I_{1}}, m_{6} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}$ $(\frac{\ddot{y}}{\dot{x} \dot{z}} - \frac{\ddot{x}}{{\dot{x}}^{2}}) = \frac{2 (a - x)}{(1 + x^{2})} - \frac{y \dot{x} (1 + a^{2})}{\dot{y} (1 + x^{2})^{2}}, \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}} = g (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}),$ $2 x + (1 + x^{2}) (\frac{\ddot{z}}{\dot{x} \dot{z}} - \frac{\ddot{x}}{{\dot{x}}^{2}}) = h (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}})$
$A_{4, 7}^{a}$	5	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{1}{(1 + {I_{2}}^{2})} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{1}{(1 + {I_{2}}^{2})} \frac{d^{2} I_{2}}{d {I_{1}}^{2}} - 2 \frac{I_{2}}{(1 + {I_{2}}^{2})^{2}} {(\frac{d I_{2}}{d I_{1}})}^{2},$ $m_{6} = \frac{1}{(1 + {I_{2}}^{2})} \frac{d^{2} I_{3}}{d {I_{1}}^{2}} - 2 \frac{I_{2}}{(1 + {I_{2}}^{2})^{2}} {(\frac{d I_{3}}{d I_{1}})}^{2}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + \frac{2 x {\dot{x}}^{2}}{(1 + x^{2})} + {\dot{x}}^{2} (1 + x^{2}) f (I_{1}, I_{3}, \frac{1}{(1 + {I_{2}}^{2})} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{1}{(1 + {I_{2}}^{2})} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 8}$	3	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 10}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{2} - \frac{{I_{1}}^{2}}{2}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}} - I_{1}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = J, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = f (I_{2} - \frac{{I_{1}}^{2}}{2}, I_{3}, \frac{d I_{2}}{d I_{1}} - I_{1}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = g (I_{2} - \frac{{I_{1}}^{2}}{2}, I_{3}, \frac{d I_{2}}{d I_{1}} - I_{1}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 10}$	5	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 11}^{a}$ $a \neq 0, 1$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = \ln (I_{1}^{\frac{1}{1 - a}}) - \frac{I_{2}}{I_{1}}, m_{2} = I_{3}, m_{3} = \ln (I_{1}^{\frac{1}{1 - a}}) + \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{J}{I_{1}^{\frac{2 - a}{a - 1}}}, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = t^{\frac{2 - a}{a - 1}} f (\ln (I_{1}^{\frac{1}{1 - a}}) - \frac{I_{2}}{I_{1}}, I_{3}, \ln (I_{1}^{\frac{1}{1 - a}}) + \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0,$ $\ddot{z} = \frac{1}{t^{2}} g (\ln (I_{1}^{\frac{1}{1 - a}}) - \frac{I_{2}}{I_{1}}, I_{3}, l n (I_{1}^{\frac{1}{1 - a}}) + \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 11}^{a}$ $a \neq 0, 1$	6	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{1}{I_{2}} \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 12}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = \frac{I_{2}}{I_{1}} + \ln (I_{1}), m_{2} = I_{3}, m_{3} = \ln (I_{1}) + \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = I_{1} J, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{1}{t} f (\frac{I_{2}}{I_{1}} + \ln (I_{1}), I_{3}, \ln (I_{1}) + \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0,$ $\ddot{z} = \frac{1}{t^{2}} h (\frac{I_{2}}{I_{1}} + \ln (I_{1}), I_{3}, \ln (I_{1}) + \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 12}$	5	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{1}{I_{2}} \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 13}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{2} - \frac{{I_{1}}^{2}}{2}, m_{2} = I_{3}, m_{3} = - I_{1} + \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = J e^{I_{1}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = e^{- t} f (I_{2} - \frac{{I_{1}}^{2}}{2}, I_{3}, - I_{1} + \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = h (I_{2} - \frac{{I_{1}}^{2}}{2}, I_{3}, - I_{1} + \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 13}$	5	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 14}^{a, b}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{2} {I_{1}}^{\frac{b - 1}{1 - a}}, m_{2} = I_{3}, m_{3} = {I_{1}}^{\frac{b - a}{1 - a}} \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = (J) {I_{1}}^{\frac{1 - 2 a}{1 - a}}, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = t^{\frac{2 a - 1}{1 - a}} f (I_{2} {I_{1}}^{\frac{b - 1}{1 - a}}, I_{3}, {I_{1}}^{\frac{b - a}{1 - a}} \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, \ddot{z} = \frac{1}{t^{2}} h (I_{2} {I_{1}}^{\frac{b - 1}{1 - a}}, I_{3}, {I_{1}}^{\frac{b - a}{1 - a}} \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 14}^{a, b}$	7	$I_{1} = z, I_{2} = e^{(a - 1) x}, I_{3} = y, \frac{d I_{2}}{d I_{1}} = (a - 1) e^{(a - 1) x} \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}},$ $\frac{d^{2} I_{2}}{d {I_{1}}^{2}} = (a - 1) e^{(a - 1) x} (\frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}} + \frac{(a - 1) {\dot{x}}^{2}}{{\dot{z}}^{2}}), \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{1}{I_{2}} \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} - (a - 1) {\dot{x}}^{2} + \frac{{\dot{z}}^{2} e^{(1 - a) x}}{(a - 1)} f (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{1}{I_{2}} \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 14}^{a, a}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = \frac{I_{2}}{I_{1}}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = (J) {I_{1}}^{\frac{1 - 2 a}{1 - a}}, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = t^{\frac{2 a - 1}{1 - a}} f (\frac{I_{2}}{I_{1}}, I_{3}, \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, t^{2} \ddot{z} = h (\frac{I_{2}}{I_{1}}, I_{3}, \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 14}^{a, 1}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{2}, m_{2} = I_{3}, m_{3} = I_{1} \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = (J) {I_{1}}^{\frac{1 - 2 a}{1 - a}}, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = t^{\frac{2 a - 1}{1 - a}} f (I_{2}, I_{3}, I_{1} \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}), \ddot{y} = 0, t^{2} \ddot{z} = h (I_{2}, I_{3}, I_{1} \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}})$
$A_{4, 15}$	2	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = I_{1}, m_{2} = I_{2}, m_{3} = I_{3}, m_{4} = \frac{d I_{2}}{d I_{1}}, m_{5} = \frac{d I_{3}}{d I_{1}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}} with condition : w = \ddot{x} = 0$ $\ddot{x} = 0, \ddot{y} = 0, \ddot{z} = h (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}})$
	4	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
	5	$I_{1} = z, I_{2} = x, I_{3} = y, \frac{d I_{2}}{d I_{1}} = \frac{\dot{x}}{\dot{z}}, \frac{d I_{3}}{d I_{1}} = \frac{\dot{y}}{\dot{z}}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \frac{\ddot{x}}{{\dot{z}}^{2}} - \frac{\dot{x} \ddot{z}}{{\dot{z}}^{3}}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \frac{\ddot{y}}{{\dot{z}}^{2}} - \frac{\dot{y} \ddot{z}}{{\dot{z}}^{3}}$ $m_{1} = I_{1}, m_{2} = I_{2}, m_{3} = I_{3}, m_{4} = \frac{d I_{2}}{d I_{1}}, m_{5} = \frac{d I_{3}}{d I_{1}}, m_{6} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{7} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = \frac{\dot{x} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} f (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{y} = \frac{\dot{y} \ddot{z}}{\dot{z}} + {\dot{z}}^{2} g (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 16}^{a, b}$ $a > 0$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : \frac{d^{2} I_{2}}{d I_{1}^{2}} = 0$ $m_{1} = {({I_{1}}^{2} + {I_{2}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{I_{2}}{I_{1}})}, m_{2} = I_{3}, m_{3} = {({I_{1}}^{2} + {I_{2}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{d I_{2}}{d I_{1}})},$ $m_{4} = \frac{{(1 + {\frac{d I_{2}}{d I_{1}}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{d I_{2}}{d I_{1}})}}{\frac{d I_{3}}{d I_{1}}}, m_{5} = J {(1 + {(\frac{d I_{2}}{d I_{1}})}^{2})}^{- 1} e^{(2 b - a) \arctan (\frac{d I_{2}}{d I_{1}})},$ $m_{6} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}} {(1 + {(\frac{d I_{2}}{d I_{1}})}^{2})}^{- 1} e^{2 (b - a) \arctan (\frac{d I_{2}}{d I_{1}})}$ $\ddot{x} = (1 + {\dot{y}}^{2}) e^{(a - 2 b) \arctan \dot{y}} f ({({I_{1}}^{2} + {I_{2}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{I_{2}}{I_{1}})}, I_{3}, {({I_{1}}^{2} + {I_{2}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{d I_{2}}{d I_{1}})}),$ $\ddot{y} = 0,$ $\ddot{z} = (1 + {\dot{y}}^{2}) e^{2 (a - b) \arctan \dot{y}} h ({({I_{1}}^{2} + {I_{2}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{I_{2}}{I_{1}})}, I_{3}, {({I_{1}}^{2} + {I_{2}}^{2})}^{\frac{1}{2}} e^{(a - b) \arctan (\frac{d I_{2}}{d I_{1}})})$
$A_{4, 18}^{b}$ $b = 0$	2	$I_{1} = t,, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \ddot{y}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : J = \ddot{x} = 0$ $m_{1} = I_{2}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = \frac{d I_{3}}{d I_{1}}, m_{5} = \frac{d^{2} I_{2}}{d {I_{1}}^{2}}, m_{6} = \frac{d^{2} I_{3}}{d {I_{1}}^{2}}$ $\ddot{x} = 0, \ddot{y} = g (I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}), \ddot{z} = h (I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}})$
$A_{4, 20}$	1	$I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \ddot{y}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition : J = \ddot{x} = 0$ $m_{1} = I_{2}, m_{2} = I_{3}, m_{3} = (1 + {I_{1}}^{2}) \frac{d I_{2}}{d I_{1}}, m_{4} = (1 + {I_{1}}^{2}) \frac{d I_{3}}{d I_{1}},$ $m_{5} = {(1 + {I_{1}}^{2})}^{2} \frac{d^{2} I_{2}}{d {I_{1}}^{2}} + 2 I_{1} (1 + {I_{1}}^{2}) \frac{d I_{2}}{d I_{1}}, m_{6} = {(1 + {I_{1}}^{2})}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}} + 2 I_{1} (1 + {I_{1}}^{2}) \frac{d I_{3}}{d I_{1}}$ $\ddot{x} = 0, \ddot{y} = - \frac{2 t \dot{y}}{(1 + t^{2})} + \frac{1}{(1 + t^{2})^{2}} g (I_{2}, I_{3}, (1 + {I_{1}}^{2}) \frac{d I_{2}}{d I_{1}}, (1 + {I_{1}}^{2}) \frac{d I_{3}}{d I_{1}}),$ $\ddot{z} = - \frac{2 t \dot{z}}{(1 + t^{2})} + \frac{1}{(1 + t^{2})^{2}} h (I_{2}, I_{3}, (1 + {I_{1}}^{2}) \frac{d I_{2}}{d I_{1}}, (1 + {I_{1}}^{2}) \frac{d I_{3}}{d I_{1}})$

Singular invariants and invariant equations

Singularity in physical systems has a key role in their analysis and often is reflected in the form of DEs in their mathematical modelling. Sometimes this singularity appears hidden in the underling physical systems and apparently not recognized at first glance in the arising DEs of the mathematical model.

We focus on a system of three second-order ODEs in the general form

\begin{matrix} \ddot{x} & = f (t, x, y, z, \dot{x}, \dot{y}, \dot{z}) \\ \ddot{y} & = g (t, x, y, z, \dot{x}, \dot{y}, \dot{z}) \\ \ddot{z} & = h (t, x, y, z, \dot{x}, \dot{y}, \dot{z}) . \end{matrix}

(1)

Our aim is to explore the singularity in the invariant structure associated with the system (1) which possess four-dimensional Lie algebras. We adopt the procedure of^3,4,20,12 in constructing the invariant structure for the system (1). For completeness, we describe some basic notions that are necessary for the sequel.

A differential function $ϕ$ is said to be invariant under the symmetry group generated by $X$ if

X^{[λ]} (ϕ) = 0,

(2)

where

X

is a symmetry vector and

X^{[λ]}

is its

λ

th prolongation.

If the rank of the coefficient matrix of vector fields on the solution manifold is less than the rank of the coefficient matrix on the generic manifold, then singularity occurs in the invariant structure of the associated underlying vector fields. This singularity has a significant role in the construction of the corresponding system of invariant DEs, which further affects their integrability. Here we investigate such type of singularity for the system (1) which admits a four-dimensional Lie algebra.

We employ the same technique discussed in literary works^3,4,20,12 We extend this to a system of three second-order ODEs given in the form (1) having Lie algebra of dimension $4$ . We explain this procedure by considering the Lie algebra $A_{4, 2}^{1}$ as given in.²⁰

Let us investigate the case of the first realization of $A_{4, 2}$ , viz.

A_{4, 2}^{1} ≅ ⟨ \partial_{x}, t \partial_{t} + x \partial_{x} + y \partial_{y}, t \partial_{x}, y \partial_{x} ⟩ .

(3)

The above realization

A_{4, 2}^{1}

has subalgebra

A_{3, 1}^{3}

. By invoking the results of,¹² we find the following set of invariants for

A_{3, 1}^{3}

I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} & \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \ddot{y} = 0.

(4)

Then by employing the prolongation of the second generator of

A_{4, 2}^{1}

on (4), we arrive at the following invariants

m_{1} = \frac{I_{2}}{I_{1}}, m_{2} = I_{3}, m_{3} = \frac{d I_{2}}{d I_{1}}, m_{4} = I_{1} \frac{d I_{3}}{d I_{1}}, m_{5} = I_{1} J, m_{6} = {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}} and \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \ddot{y} = 0.

(5)

The associated invariant system of ODEs for (5) then is

\begin{matrix} I_{1} J & = f (\frac{I_{2}}{I_{1}}, I_{3}, \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}) \\ \frac{d^{2} I_{2}}{d {I_{1}}^{2}} & = 0 \\ {I_{1}}^{2} \frac{d^{2} I_{3}}{d {I_{1}}^{2}} & = h (\frac{I_{2}}{I_{1}}, I_{3}, \frac{d I_{2}}{d I_{1}}, I_{1} \frac{d I_{3}}{d I_{1}}) \end{matrix}

(6)

In similar manner, the corresponding system of three second-order ODEs for (5) is

\begin{matrix} t \ddot{x} & = f (\frac{y}{t}, z, \dot{y}, t \dot{z}) \\ \ddot{y} & = 0 \\ t^{2} \ddot{z} & = h (\frac{y}{t}, z, \dot{y}, t \dot{z}) . \end{matrix}

(7)

By implementing the procedure as in the work of Ayub et al.,²⁰ it is found that by invoking (7), the rank of the solution manifold becomes less than the rank of the generic manifold for

A_{4, 2}^{1}

. Hence, the underlying invariants (5) and associated invariant system admitting

A_{4, 2}^{1}

have singularity.

By utilizing the realizations of Lie algebras of dimension $4$ presented in,^20,19 we classify such type of singularity in the corresponding differential invariants for the system (1) having Lie algebras of dimension $4$ . A list of such possible singular differential invariants in (1+3-dimensional space and the corresponding canonical forms admitting the Lie algebras of dimension $4$ are provided in Table 1 given at the end.

The singularity structure and corresponding invariants for four-dimensional Lie algebras in (1+3)-dimensional space can be categorized into different types and this is discussed in the next section.

Types of singular invariant equations

Singularity in physical systems may arise in different situations with various physical constraints as a result of which there are different types. So it is natural that diverse types of singularity would be reflected in the DEs associated with the mathematical modelling of these physical systems. There are some physical systems in which singularity depends on some constraints in the form of conditions and known as the conditional singularity.

If we look at the conditional singularity from a mathematical point of view in the case of a system of three second-order ODEs having Lie algebras of dimension $4$ , then one sees that the Lie algebra $A_{4, 1}^{2}$ leads to conditional singularity in the associated invariants.

For the case of a system of three second-order ODEs possessing the Lie algebra $A_{4, 1}^{2}$ , by following similar analysis as performed in the previous section, we deduce the singular invariants

I_{1} = t, I_{2} = y, I_{3} = z, \frac{d I_{2}}{d I_{1}} = \dot{y}, J = \ddot{x}, \frac{d I_{3}}{d I_{1}} = \dot{z}, \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = \ddot{z} with condition \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = \ddot{y} = 0

(8)

with the extra differential condition

f_{I_{1} I_{1}} + 2 \frac{d I_{2}}{d I_{1}} f_{I_{1} I_{2}} + {(\frac{d I_{2}}{d I_{1}})}^{2} f_{I_{2} I_{2}} = 0

(9)

and the associated conditional invariant system of ODEs

\begin{matrix} J = k (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}) \\ \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = 0 \\ \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = h (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}) . \end{matrix}

(10)

By employing (8) in (10), the corresponding system of three second-order ODEs possessing the Lie algebra

A_{4, 1}^{2}

with the extra condition (9) can be found.

In a similar manner as performed for the case of conditional singularity, on the basis of other properties of the underlying physical systems such as weak uncoupling, weak linearity, partially uncoupling and partial linearity, singularity in differential invariants of these investigated systems from an algebraic point of view can be classified. This classification can be utilized in order to investigate the singularity analysis of the underlying physical systems as an inverse problem.

We classify singularity in differential invariants and the associated canonical forms, mentioned in Table 1, on the basis of weak uncoupling, partially uncoupling, weak linearity and partially linearity in 1+3-dimensional space for Lie algebras of dimension $4$ .

In the light of the above observations, we proceed to the following definitions which are extensions of what appears in the work of Ayub et al.³

Definition 1

A system of three second-order ODEs is said to be

weakly uncoupled if one of its equations possesses an uncoupled the scalar second-order equation in one of the dependent variables;

partially uncoupled if two of its equations consist of uncoupled scalar second-order equations in their own right in two of their dependent variables.

In a similar manner, we can define the linearity of a system of three second-order ODEs in terms of weakly linear and partially linear as follows.

Definition 2

A system of three second-order ODEs is said to be

weakly linear if one of its equations constitute a scalar linear second-order equation;

partially linear if two of its equations embody scalar linear second-order equations.

Furthermore, we also define weakly linearizable and partially linearizable systems of three second-order ODEs as below.

Definition 3

A system of three second-order ODEs is said to be

weakly linearizable by point transformation if it can be transformed via an invertible transformation to a weakly linear form;

partially linearizable via point transformation if it can be transformed by an invertible transformation to a partially linear form.

On the basis of the above definitions, the classification results associated with canonical forms presented in Table 1 can be summarized as follows.

Proposition 1

A system of three second-order ODEs given in terms of singular invariant equations is said to be weakly uncoupled and weakly linearizable by point transformation if and only if it possesses one of the following Lie algebras

\begin{matrix} A_{4, 2}^{1}, A_{4, 3}^{1}, A_{4, 4}^{2}, A_{4, 5}^{1}, A_{4, 6}^{(1), 1}, A_{4, 6}^{(1), 4} (ψ^{'} (x) = 0), A_{4, 6}^{(a), 1}, A_{4, 7}^{(a), 4}, A_{4, 10}^{1}, A_{4, 11}^{(a), 1}, A_{4, 12}^{1}, A_{4, 13}^{1}, \\ A_{4, 14}^{(a, b), 1}, A_{4, 14}^{(a, a), 1}, A_{4, 14}^{(a, 1), 1}, A_{4, 16}^{(a, b), 1}, A_{4, 18}^{(b), 2} (b = 0), A_{4, 20}^{1} with rank r = 2 in all cases . \end{matrix}

(11)

Proposition 2

A system of three second-order ODEs represented in terms of singular invariant equations is said to be partially uncoupled and partially linearizable via point transformation if and only if it possesses the Lie algebra $A_{4, 15}^{2}$ with rank $r = 1$ .

Proposition 3

A system of three second-order ODEs written in terms of singular invariant equations can be reduced to asystem of two second-order ODEs by point transformation if and only if it admits one of the following Lie algebras

\begin{aligned} A_{4, 5}^{6}, A_{4, 7}^{(a), 5}, A_{4, 8}^{3}, A_{4, 10}^{5}, A_{4, 11}^{(a), 6}, A_{4, 12}^{5}, A_{4, 13}^{5}, A_{4, 14}^{(a, b), 7}, A_{4, 15}^{4} with rank r = 2 \\ or A_{4, 1}^{3}, A_{4, 15}^{5} with rank r = 1 or A_{4, 6}^{(a), 7} with rank r = 3. \end{aligned}

Proposition 4

The set of following realizations in 1+3-dimensional space associated with four-dimensional Lie algebras, viz.

A_{4, 1}^{3}, A_{4, 5}^{6}, A_{4, 6}^{(a), 7}, A_{4, 7}^{(a), 5}, A_{4, 8}^{3}, A_{4, 10}^{5}, A_{4, 11}^{(a), 6}, A_{4, 12}^{5}, A_{4, 13}^{5}, A_{4, 14}^{(a, b), 7}, A_{4, 15}^{4}, A_{4, 15}^{5}

do not lead to a complete system of three second-order ODEs.

In a similar manner for conditional singularity for differential invariants in 1+3-dimensional space for a Lie algebra of dimension $4$ , the results on the classification given in Table 1 can be summarized as below.

Proposition 5

‘A system of three second-order ODEs represented in terms of conditional singular invariant equations by point transformation if and only if it admits the Lie algebra $A_{4, 1}^{2}$ with rank $r = 1$ provided that $f_{t t} + 2 \frac{d y}{d t} f_{t y} + (\frac{d y}{d t})^{2} f_{y y} = 0.$ ’

Proposition 6

‘A system of three second-order ODEs represented in terms of conditional singular invariant equations is said to be conditional weakly uncoupled and conditional weakly linearizable by point transformation if and only if it possesses the Lie algebra $A_{4, 1}^{2}$ with rank $r = 1$ provided that $f_{t t} + 2 \frac{d y}{d t} f_{t y} + (\frac{d y}{d t})^{2} f_{y y} = 0.$ ’

The proofs of these propositions follow directly from Table 1.

Integration

Lie point symmetries are utilized for various types of analyses of DEs. One of the important features is the reduction and integrability of the underlying DEs. Especially, in the case of a system of DEs, a reduction procedure in the classical manner is not effective.¹⁸ Therefore, for integrability of a system of DEs, the canonical form approach is of much benefit. This is equally applicable in both cases, i.e. for regular and singular invariant system of equations. Here, we consider the singular invariant system of equations. The singular differential invariant structure and construction of canonical forms is described in the “Singular invariants and invariant equations” section. In addition, categorization of the classified forms as stated in Table 1 has been investigated in detail in the “Types of singular invariant equations” section-(2.1).

We visit some cases from Table 1 and perform the integrability analysis of the underlying singular invariant equations for peculiar values of $f, g, h$ , and $k$ . On the basis of the categorization described in Table 1, we delve into the integrability analysis of these classified canonical forms for the following three categories.

Category A

Those cases of canonical forms are analysed for which $\ddot{y} = 0$ apply. The admitting Lie algebra of dimension $4$ for the system of three second-order ODEs has the Lie subalgebra $A_{3.1}^{3}$ .

Let us consider a particular class of a Lie algebra from Category A viz., $A_{4, 14}^{(a, a), 1}$ given in Table 1 which has the Lie subalgebra $A_{3.1}^{3}$ and the corresponding singular invariant system is given by
$\begin{matrix} \ddot{x} & = t^{2 a - 1 / 1 - a} f (\frac{y}{t}, z, \dot{y}, t \dot{z}) \\ \ddot{y} & = 0 \\ t^{2} \ddot{z} & = h (\frac{y}{t}, z, \dot{y}, t \dot{z}) . \end{matrix}$
(12)

Here we look at the following two cases.
Trivial case $f = h = c o n s t a n t$ .

When $h$ is nonconstant.
(a) Trivial case $f = h = c o n s t a n t s a y f = h = 1$

The system (12) has a solution
$\begin{aligned} x & = \frac{(1 - a)^{2}}{a} t^{\frac{1}{1 - a}} + b_{1} t + b_{2}, \\ y & = b_{3} t + b_{4}, \\ z & = - \ln t + b_{5} t + b_{6}, \end{aligned}$
where $b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, and b_{6}$ are arbitrary constants.

(b) When $h$ is nonconstant

If we take the value of $h (\frac{y}{t}, z, \dot{y}, t \dot{z}) = \frac{y}{t} - \dot{y} - t \dot{z} + z$ , then the system (12) takes the form
$\begin{matrix} \ddot{x} & = t^{2 a - 1 / 1 - a} f (\frac{y}{t}, z, \dot{y}, t \dot{z}) \\ \ddot{y} & = 0 \\ t^{2} \ddot{z} & = \frac{y}{t} - \dot{y} - t \dot{z} + z . \end{matrix}$
(13)
The solution of the system (13) has the form
$\begin{aligned} x & = \int (\int t^{2 a - 1 / 1 - a} f (d_{1} + \frac{d_{2}}{t}, \frac{d_{3}}{t} + d_{4} t - \frac{d_{2}}{2 t} \ln t, d_{1}, - \frac{(d_{2} + 2 d_{3})}{t} + d_{4} t + \frac{d_{2}}{2 t} \ln t) d t) \\ d t + d_{5} t + d_{6}, y = d_{1} t + d_{2}, z = \frac{d_{3}}{t} + d_{4} t - d_{2} \frac{\ln t}{2 t}, \end{aligned}$
where $d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, and d_{6}$ are arbitrary constants.

In both Cases (a) and (b) of the studied Lie algebra $A_{4, 14}^{(a, a), 1}$ , the corresponding invariant system is integrable. If we look further in the integrability for this algebra $A_{4, 14}^{(a, a), 1}$ , we find that the second equation incorporates integrability of the remaining two equations of the system (12). Moreover, we also obtain that the underlying associated invariant system is integrable when the value of $h$ is such that the third equation of the system (12) becomes solvable for $z$ .

Proceeding in like manner for all Lie algebras in Category A, as mentioned in Table 1, we deduce that the integrability of these corresponding invariant systems depends on a scalar second-order ODE associated with this investigated invariant system.

Category B

In this category, those cases of canonical forms are analysed in which $\ddot{x} = 0$ is included. Moreover, in this case, the possessing Lie algebra of dimension $4$ of the system of three second-order ODEs has a Lie subalgebra $A_{3.5}^{3}$ .

Consider the Lie algebra taken from Category B viz., $A_{4, 20}^{1}$ as given in Table 1 which possesses the Lie subalgebra $A_{3.5}^{3}$ . The associated constructed singular invariant equations are given by
$\begin{matrix} \ddot{x} & = 0 \\ \ddot{y} & = \frac{- 2 t \dot{y}}{1 + t^{2}} + \frac{1}{(1 + t^{2})^{2}} g (y, z, (1 + t^{2}) \dot{y}, (1 + t^{2}) \dot{z}) \\ \ddot{z} & = \frac{- 2 t \dot{z}}{1 + t^{2}} + \frac{1}{(1 + t^{2})^{2}} h (y, z, (1 + t^{2}) \dot{y}, (1 + t^{2}) \dot{z}) . \end{matrix}$
(14)

Like the previous Category A, we study the following two cases for this underlying class $A_{4, 20}^{1}$ .
Trivial case $g = h = constant$ .

When $g$ and $h$ are nonconstant.
(a) Trivial case $g = h = constant s a y g = h = 1$

In particular, if we take $g = h = 1$ , then the above singular invariant system assumes the form
$\begin{matrix} \ddot{x} & = 0, \\ \ddot{y} & = \frac{- 2 t \dot{y}}{1 + t^{2}} + \frac{1}{(1 + t^{2})^{2}}, \\ \ddot{z} & = \frac{- 2 t \dot{z}}{1 + t^{2}} + \frac{1}{(1 + t^{2})^{2}} . \end{matrix}$
(15)
Solving system (15) simultaneously, we get the solution of this invariant system as
$\begin{aligned} x & = A t + B, \\ y & = \frac{1}{2} (\arctan t)^{2} + C \arctan t + D, \\ z & = \frac{1}{2} (\arctan t)^{2} + E \arctan t + F . \end{aligned}$
Proceeding in a similar manner as in the previous Category A, for the Category B, we observe that when $g$ and $h$ are nonconstant, then the first equation does not incorporate in the integrability of the remaining two equations of the system (14). Moreover, we find that the system (14) is integrable when the system of two remaining equations become solvable. In other words, the integrability of the system for this Lie algebra $A_{4, 20}^{1}$ , depends on the system of two second-order ODEs associated with the corresponding singular invariant system which admits this Lie algebra.

Extending this analysis to all the Lie algebras appearing in Category B, in Table 1, we determine that the integrability of these corresponding invariant systems having the underlying Lie algebras, depends on a system of two scalar second-order ODEs associated with their invariant systems.

Category C

Those cases of canonical forms are analysed here in which both $\ddot{x} = 0$ and $\ddot{y} = 0$ are considered.

Consider the Lie algebra $A_{4.15}^{2}$ presented in Table 1. The invariant representation of the system is
$\begin{matrix} J = 0 \\ \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = 0 \\ \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = f (I_{1}, I_{2}, I_{3}, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}) . \end{matrix}$
(16)

The above relations are as shown in Table 1 and we determine
$\begin{matrix} \ddot{x} & = 0 \\ \ddot{y} & = 0 \\ \ddot{z} & = f (t, y, z, \dot{y}, \dot{z}) . \end{matrix}$
(17)
Here the first two equations of the invariant system (17) incorporate into the integrability of the third equation. After simple calculation, the above system results in
$\begin{matrix} x & = c_{1} t + c_{2}, \\ y & = c_{3} t + c_{4}, \\ \ddot{z} & = f (t, c_{3} t + c_{4}, z, c_{3}, \dot{z}), \end{matrix}$
(18)
which can be solved for any particular value of $f$ and arbitrary constants $c_{3}, c_{4}$ . The corresponding invariant system (17) is integrable if the third equation of system (18) becomes solvable. Thus the integrability of the system with Lie algebra $A_{4.15}^{2}$ depends upon the scalar second-order ODE of the corresponding invariant system.

In the light of the above discussions, integrability analysis here can be summarized as follows.
Proposition 7
If a system of three second-order ODEs when represented in terms of singular invariant equations possess a four-dimensional solvable symmetry Lie algebra, then its general solution by quadrature exists in the following two cases.

Case (1). If the admitting Lie algebra is $A_{4.15}^{2}$ or it possesses a Lie subalgebra $A_{3.1}^{3}$ , then by utilization of point transformation, its general solution by quadrature depends on the solution of a scalar second-order ODE.

Case (2). If the admitting Lie algebra possesses a Lie subalgebra $A_{3.5}^{3}$ , then by utilization of point transformation, its general solution by quadrature depends on the solution of a system of two second-order ODEs.

The proofs of these results follow directly from Table 1. The Case 1 is analysed in Categories A and C and Case 2 is discussed in Category B.

Applications

In many physical systems such as vibratory and oscillatory systems, the energy in the form of the intensity of vibrations or oscillations is gradually dissipated by friction and other resistances with time; these are known as damped. In mathematical modelling, they play a crucial role in the mathematical study of mechanics of the system of interacting as well as non-interacting particles. Singularity in the invariant structure of such types of physical systems is a peculiar hidden structure which is closely related to various aspects of analysis including damping and integrability. In the current study, we investigate the classification of singularity in invariant structure for a four-dimensional Lie algebra and have presented the results in the form of a table.

In addition, one of the interesting properties related to a solution of DEs is the notion of blow-up of the corresponding solution in finite time. Here we briefly mention this aspect.

Blow-up of a solution in finite time:

The blow-up of a solution of the underlying DE in finite time indicates the phenomenon that the corresponding solution tends to infinity in a finite time. The study concerning the blow-up phenomenon is evoked by fruitful mathematical structures of the investigated equation. In addition, the blow-up mechanisms have special meaning emphasizing physical and biological significances in applications. Mathematically, let
$L [u (t)] = G (t)$
(19)
be a DE with $L$ a differential operator, $G (t)$ some function of $t$ and say $u (t) = ϕ (t)$ defined in an open interval $I D \subset I R$ a solution of (19). If
${lim}_{t \to T} ϕ (t) \to \pm \infty,$
(20)
for $T \in I D$ , then $u (t)$ becomes infinite in finite time $T$ and consequently the blow-up of the solution occurs as $t \to T$ .

If the DE is autonomous, the independent variable, time $t$ , can be moved by time translation to shift the blow-up time. In the context of this work, this condition arises if the system has translation symmetry in $t$ .

In this section, we present two physical examples from mechanics to illustrate our main results.

I. We consider the partially coupled system of three particles possessing small oscillations governed by the system
$\begin{matrix} \ddot{x} & = - k_{1} {\dot{z}}^{2} - k_{2} k_{1} t z \dot{y} e^{k_{1} y} \\ \ddot{y} & = - k_{1} {\dot{y}}^{2} \\ \ddot{z} & = k_{1} \frac{\dot{y} \dot{z} e^{k_{1} y}}{t} - k_{1} \frac{z \dot{y} e^{k_{1} y}}{t^{2}}, t > 0 \end{matrix}$
(21)
in a small region; $a_{1} < x < a_{2}, b_{1} < y < b_{2} and c_{1} < z < c_{2}$ such that for any small number $ϵ > 0,$ $| a_{1} - a_{2} |^{2} = ϵ, | b_{1} - b_{2} |^{2} = ϵ and | c_{1} - c_{2} |^{2} = ϵ$ , where $ϵ \to 0$ . Due to small oscillations, ranges of effective variables have been taken small and also gravity effects are negligible. In addition, $k_{1}$ and $k_{2}$ are arbitrary constants possessing dimensions $L^{- 1}$ and $L^{- 1} T^{- 2}$ , respectively. During their motion, there is resistance which is described by the second particle facing the drag resistance that is equal to the square of its velocity. The other two particles have time-varying resistance, i.e. they depend on the velocities of the second and third particles as well as also on the value of the third particle.

The system (21) admits the symmetry Lie algebra of vector fields
$\partial_{x}, \frac{e^{- k_{1} y}}{k_{1}} \partial_{y}, e^{k_{1} y} \partial_{x} + \frac{μ (t) e^{- k_{1} y}}{k_{1}} \partial_{y}, t \partial_{x} .$
(22)
By invertible transformations
$\bar{t} = t, \bar{x} = x, \bar{y} = \frac{1}{k_{1}} \ln y, \bar{z} = z,$
(23)
we determine that the transformed vectors become the specific case of the realization $A_{4, 4}^{2}$ as given in.⁴ In addition, the nonlinear system (21) is transformed to the canonical form, after dropping bars,
$\begin{matrix} \ddot{x} & = - k_{1} {\dot{z}}^{2} - k_{2} t z \dot{y} \\ \ddot{y} & = 0 \\ \ddot{z} & = k_{1} \frac{\dot{y} \dot{z}}{t} - k_{1} \frac{z \dot{y}}{t^{2}} . \end{matrix}$
(24)
By employing invariants of Table 1, for the underlying case of realization $A_{4, 4}^{2}$ , we deduce that the invariant representation of (24) has the form
$\begin{matrix} J = - k_{1} {(\frac{d I_{3}}{d I_{1}})}^{2} - k_{2} I_{1} \frac{d I_{2}}{d I_{1}} \\ \frac{d^{2} I_{2}}{d {I_{1}}^{2}} = 0 \\ \frac{d^{2} I_{3}}{d {I_{1}}^{2}} = - k_{1} \frac{1}{I_{1}} \frac{d I_{3}}{d I_{1}} \frac{d I_{2}}{d I_{1}} - k_{1} \frac{I_{3}}{{I_{1}}^{2}} \frac{d I_{2}}{d I_{1}} \end{matrix}$
(25)
The algebra $A_{4, 4}^{2}$ has a Lie subalgebra $A_{3.1}^{3}$ and thus lies in Category A and its integrability is according to Proposition 7. Hence we follow the procedure described in Category A.

From the second equation of the system (25), we arrive at
$I_{2} = d_{1} I_{1} + d_{2},$
(26)
where $d_{1}, d_{2}$ are constants of integration. Performing some manipulations, we obtain the solution of the third equation in the system (25) in the form
$I_{3} = t^{d_{1} k_{1}} d_{4} + \frac{t d_{3}}{1 - d_{1} k_{1}},$
(27)
where $d_{3}, d_{4}$ are constants of integration.

By using the values of (27) in the first equation of the system (25) and performing some manipulations, we obtain
$\begin{aligned} x & = \frac{t^{2 d_{1} k_{1}} d_{1} d_{4}^{2} k_{1}^{2}}{2 (1 - 2 d_{1} k_{1})} - \frac{t^{1 + d_{1} k_{1}} d_{4} (- 2 d_{3} k_{1} (2 + d_{1} k_{1}) (3 + d_{1} k_{1}) + t^{2} d_{1} (- 1 + d_{1}^{2} k_{1}^{2}) k_{2})}{(- 1 + d_{1}^{2} k_{1}^{2}) (2 + d_{1} k_{1}) (3 + d_{1} k_{1})} \\ + \frac{t^{2} d_{3} (- 6 d_{3} k_{1} + t^{2} d_{1} (- 1 + d_{1} k_{1}) k_{2})}{12 (- 1 + d_{1} k_{1})^{2}} + d_{5} I_{1} + d_{6}, \end{aligned}$
(28)
where $d_{5}, d_{6}$ are constants of integration with holding conditions $d_{1} \neq \pm \frac{1}{k_{1}}, \frac{1}{2 k_{1}}, - \frac{2}{k_{1}}, - \frac{3}{k_{1}}$ .

By utilizing invariants in Table 1 and employing invertible transformation (23), reverting to the original variables after dropping the bars, (26), (27) and (28) become
$\begin{aligned} x & = \frac{t^{2 d_{1} k_{1}} d_{1} d_{4}^{2} k_{1}^{2}}{2 (1 - 2 d_{1} k_{1})} - \frac{t^{1 + d_{1} k_{1}} d_{4} (- 2 d_{3} k_{1} (2 + d_{1} k_{1}) (3 + d_{1} k_{1}) + t^{2} d_{1} (- 1 + d_{1}^{2} k_{1}^{2}) k_{2})}{(- 1 + d_{1}^{2} k_{1}^{2}) (2 + d_{1} k_{1}) (3 + d_{1} k_{1})} \\ + \frac{t^{2} d_{3} (- 6 d_{3} k_{1} + t^{2} d_{1} (- 1 + d_{1} k_{1}) k_{2})}{12 (- 1 + d_{1} k_{1})^{2}} + d_{5} I_{1} + d_{6}, \end{aligned}$
(29)
$y = \ln (d_{1} t + d_{2})$
(30)
$z = t^{d_{1} k_{1}} d_{4} + \frac{t d_{3}}{1 - d_{1} k_{1}},$
(31)
where $d_{1}, d_{2}, d_{3}, d_{4}, d_{5}$ and $d_{6}$ are constants of integration providing that $d_{1} \neq \pm \frac{1}{k_{1}}, \frac{1}{2 k_{1}}, - \frac{2}{k_{1}}, - \frac{3}{k_{1}}$ .

Thus (29)–(31) provide the solution of our system (21).

We now briefly comment on the blow-up of the solution above. By careful observation, the solution, viz., (29)–(31) of the system (21), results in the blow-up of the solution in time $t$ occurs as $d_{1} t + d_{2} \to 0$ provided that $d_{1}$ and $d_{2}$ have opposite signs.

II. Let us now consider the motion of a system of three interacting particles in a resistive medium ruled by the system
$\begin{matrix} \ddot{x} & = - \dot{x} k_{1} ({\dot{x}}^{2} + {\dot{y}}^{2}) \\ \ddot{y} & = - \dot{y} k_{1} ({\dot{x}}^{2} + {\dot{y}}^{2}) \\ \ddot{z} & = \dot{z} k_{1} ({\dot{x}}^{2} + {\dot{y}}^{2}) + ({\dot{x}}^{2} + {\dot{y}}^{2}) k_{2} e^{- b \arctan \frac{\dot{y}}{\dot{x}}} \end{matrix}$
(32)
which like the previous problem, is also confined in a small region; $a_{1} < x < a_{2}, b_{1} < y < b_{2} and c_{1} < z < c_{2}$ such that for any small number $ϵ > 0,$ $| a_{1} - a_{2} |^{2} = ϵ, | b_{1} - b_{2} |^{2} = ϵ and | c_{1} - c_{2} |^{2} = ϵ$ , where $ϵ \to 0$ and $k_{1}, k_{2}$ are arbitrary constants whose dimension are $L^{- 2} T$ and $L^{- 1}$ . Here each particle is facing a resistance that is equal to the product of the velocity of corresponding particle and sum of squares of velocities of first two particles. In addition, a factor $({\dot{x}}^{2} + {\dot{y}}^{2}) k_{2} e^{- b \arctan \frac{\dot{y}}{\dot{x}}}$ depending on the velocities of first two ones is attached with the equation of the third particle which reduces the damping effect.

The system (32) possesses the Lie algebra spanned by the following symmetry vectors
$\partial_{t}, x \partial_{t}, y \partial_{t}, 2 b t \partial_{t} + (b x + y) \partial_{x} + (b y - x) \partial_{y} .$
(33)
By utilization of the following invertible transformations
$\bar{t} = x, \bar{x} = t, \bar{y} = y, \bar{z} = z,$
(34)
we deduce the peculiar case of the realization of the Lie algebra $A_{4, 16}^{2 b, b, 1}$ presented in the work of Ayub et al..⁴ Consequently, the nonlinear system (32) is converted into a particular case of the invariant system associated with $A_{4, 16}^{2 b, b, 1}$ presented in Table 1. We arrive at the following canonical form
$\begin{matrix} \ddot{\bar{x}} & = (1 + {\dot{\bar{y}}}^{2}) k_{1} \\ \ddot{\bar{y}} & = 0 \\ \ddot{\bar{z}} & = (1 + {\dot{\bar{y}}}^{2}) k_{2} e^{- b \arctan \dot{\bar{y}}} . \end{matrix}$
(35)
Starting from the 2nd equation of (35), we find the solution of system (35) and reverting to the original variables via transformations (34), we deduce that
$t = (1 + c_{1}^{2}) k_{1} \frac{x^{2}}{2} + c_{3} x + c_{4}$
(36)
$y = c_{1} x + c_{2} .$
(37)
$z = (1 + c_{1}^{2}) k_{2} e^{- b \arctan c_{1}} \frac{x^{2}}{2} + c_{5} x + c_{6},$
(38)
where $c_{1}, c_{2}, c_{3}, c_{4}, c_{5}$ , and $c_{6}$ are arbitrary constants and (36)–(38) form the solution of (32).

Conclusion

We gave a description of the classification problem of singularity in differential invariant structures in $(1 + 3)$ -dimensional space from an algebraic point of view and the corresponding singular invariant equations for a system of three second-order ODEs having Lie algebras of dimension 4. It has been shown that there exist different classes of singularity in the differential invariant structure and their corresponding invariant DEs.

We have obtained $20$ cases of canonical forms for Lie algebras of dimension $4$ associated with a system of three second-order ODEs in $(1 + 3)$ -dimensional space having singularity in the differential invariant structure. Of these, $18$ cases of the canonical forms in $(1 + 3)$ -dimensional space are those which can be reducible to weakly uncoupled and weakly linearizable systems by point transformations for the admitting Lie algebras of dimension $4$ . In addition, one case of canonical form $A_{4, 15}^{2}$ leads to a partially uncoupled and partially linearizable system of three second-order ODEs by point transformation in $(1 + 3)$ -dimensional space having Lie algebra of dimension $4$ . One case for a system of three second-order ODEs having Lie algebra of dimension four, viz., $A_{4, 1}^{2}$ is reducible to a conditional singular invariant system by point transformations.

In some cases due to singular invariant structure, these classified canonical forms do not lead to a complete system of three second-order ODEs in $(1 + 3)$ -dimensional space possessing a Lie algebra of dimension $4$ . There are $12$ such cases which do not lead to a system of three second-order ODEs in $(1 + 3)$ -dimensional space as mentioned in Table 1.

Furthermore, on the basis of categorization of singular invariant equations for a system of three second-order ODEs possessing Lie algebras of dimension $4$ , we have described integrability in a detailed manner for the underlying systems. We have developed an integration algorithm for these classified canonical forms as presented in Table 1, mainly in three categories, viz. Category A, Category B, Category C as elaborated. In addition, integrability aspects of some cases have been described in which particular values of functions are utilized in their construction. Two examples from mechanics of particles are presented to illustrate the utilization of the singularity in the differential invariant structure.

This classification reveals many features of singularity in differential invariants and related invariant systems of equations. This work will have significance for the classification and integrability of a system of three second-order ODEs possessing Lie algebras of dimension $4$ . Moreover, it can as well be utilized for the classification and integrability of higher-order ODEs having finite-dimensional Lie algebras.

By employing mathematical modelling of physical systems, this classification of singularity in differential invariants in $(1 + 3)$ -dimensional space possessing a Lie algebra of dimension $4$ can be used for the analysis of aspects on singularity of the physical systems as an inverse problem.

Remarks on Table 1

$f, g, h, k, and ψ$ are arbitrary functions with specified conditions as stated in the corresponding realization.

$I_{1}, I_{2}, I_{3}, J, \frac{d I_{2}}{d I_{1}}, \frac{d I_{3}}{d I_{1}}$ , and $\frac{d^{2} I_{2}}{d {I_{1}}^{2}}$ form a set of differential invariants for the associated three-dimensional subalgebras³ in each case of the Lie algebra of dimension $4$ discussed.”

$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ , and $m_{6}$ form a set of second-order singular differential invariants for the corresponding system of three second-order ODEs which admit Lie algebras of dimension $4$ in each case.

Footnotes

Acknowledgments

We appreciate the useful remarks of the reviewers and are thankful to them for such guidance by which the paper has been significantly improved.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Muhammad Ayub

Author Biographies

Dr. Muhammad Ayub got his PhD degree from Quid-e-Azam University Islamabad in 2012. Currently he is working as an Assistant Professor at COMSATS University Islamabad (Abbottabad Campus). His area of research is Symmetry Analysis of Differential Equations; particularly, differential invariants, conservation laws, Lie algebra, system of differential equations, similarity solutions, linearization of differential equations, Symmetry Analysis of Fractional Differential Equation, computational approaches of symmetry analysis and applications of symmetry analysis in different fields of applied sciences.

Zahida Sultan was born in the village Nagdar Azad Kashmir Pakistan. She is currently working in Fatimah Jinnah Post Graduate College for Women as Assistant professor in Muzaffarabad, Pakistan. She got her M.Phil degree from university of Azad Jammu and Kashmir and currently doing PhD from there also. Her research area is Symmetry Analysis of Differential Equations especially on symmetries and algebraic properties of system of ordinary differential equations.

Prof. Dr. Muhammad Naeem Qureshi got PhD degree in Applied Mathematics from Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan in 2002. He served as a Professor of Mathematics in the Department of Mathematics, University of Azad Jammu and Kashmir. His area of research includes geometry, dynamical systems, algebra and symmetry analysis of differential equations. During his carrier, he produced number of M.Phil and PhD students. Moreover on his research domain, he has published numbers of research articles in well-reputed international Journals.

Prof. Dr. Fazal Mahmood Mahomed was born in Durban, South Africa. He obtained his PhD at the University of the Witwatersrand South Africa. He is the research Professor and Director of the DSI-NRF Centre for Excellence in Mathematical and Statistical Sciences at the University of the Witwatersrand, Johannesburg, South Africa. He is a well-known and well-established researcher on algebraic, geometric and operator methods for ordinary and partial differential equations, and has published several seminal articles in the area.

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Classification of singular differential invariants in ( 1 + 3 )-dimensional space and integrability

Abstract

Keywords

Introduction

Notation

Singular invariants and invariant equations

Types of singular invariant equations

Integration

Category A

Category B

Category C

Applications

Conclusion

Remarks on Table 1

Footnotes

Acknowledgments

Declaration of conflicting interests

Funding

ORCID iD

Author Biographies

References