# Triangular Schlesinger systems and superelliptic curves

## TL;DR

This paper explores special solutions to the Schlesinger system with triangular matrices and rational eigenvalue differences, linking them to superelliptic curves and deriving explicit solutions for Painlevé VI and Garnier systems.

## Contribution

It introduces a novel approach connecting Schlesinger systems with superelliptic curves, providing explicit polynomial, rational, and algebraic solutions for these integrable systems.

## Key findings

- Solutions expressed via periods of meromorphic differentials on superelliptic curves.
- Explicit polynomial and rational solutions for specific eigenvalue differences.
- New algebraic solutions for Garnier systems.

## Abstract

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size $(p\times p)$ 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 $(2\times2)$-case, we obtain explicit sequences of rational solutions and one-parameter families of rational solutions of Painlev\'e VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.