Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients

Abstract

We consider deformations of $2\times 2$ and $3\times 3$ matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which don't satisfy the well-known system of Schlesinger equations (or its natural generalization). Some general statements concerning reducibility of such deformations for $2\times 2$ ODEs are proved. An explicit example of the general non-Schlesinger deformation of $2\times 2$-matrix ODE of the Fuchsian type with 4 singular points is constructed and application of such deformations to the construction of special solutions of the corresponding Schlesinger systems is discussed. Some examples of isomonodromy and non-isomonodromy deformations of $3\times 3$ matrix ODEs are considered. The latter arise as the compatibility conditions with linear ODEs with non-singlevalued coefficients.

15 pages, to appear in J. Phys. A

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