On the meromorphic equivalence of standard matrix differential equations

Abstract

In this paper we address the problem of determining all the meromorphic differential equations that are meromorphically equivalent to a given equation; and we also discuss how one can effectively check whether two given equations are equivalent.

In the case of equations of block size 1, which is a generalization of the case of distinct eigenvalues, we provide solutions to these problems which involve only algebraic processes. This allows a detailed description of the structure of the involved transformations. Our discussion is based on the concept of direct transformations which we define and study thoroughly.