Solving triangular Schlesinger systems via periods of meromorphic differentials

We study the Schlesinger system of PDEs for $N$ matrices of size $p\times p$ in the case when they 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. As an application to the $2\times2$-case, explicit solutions of Painleve VI equations and Garnier systems are obtained.