Triangular solutions to a Riemann-Hilbert problem from superelliptic curves

We give fundamental solutions of arbitrarily sized matrix Fuchsian linear systems, in the case where the coefficients B⁽ⁱ⁾ of the systems are matrix solutions of the Schlesinger system that are upper triangular, and whose eigenvalues follow an arithmetic progression of a rational difference. The values on the superdiagonals of the matrices B⁽ⁱ⁾ are given by contour integrals of meromorphic differentials defined on Riemann surfaces obtained by compactification of superelliptic curves.