On the Reductions and Classical Solutions of the Schlesinger equations

Abstract

The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of ``more complicated&#39;&#39; Schlesinger equations $S_{(n,m)}$ to ``simpler&#39;&#39; $S_{(n&#39;,m&#39;)}$ having $n&#39;< n$ or $m&#39; < m$.