Triangular Schlesinger systems and superelliptic curves

Abstract

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size (p×p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2×2)-case, we obtain explicit sequences of rational solutions and of one-parameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.