Note on algebro-geometric solutions to triangular Schlesinger systems

TL;DR

This paper constructs explicit algebro-geometric solutions to rank two Schlesinger systems and derives new families of solutions to the sixth Painlevé equation using elliptic curve periods, expanding the solution space.

Contribution

It introduces novel algebro-geometric upper triangular solutions to Schlesinger systems and derives explicit Painlevé VI solutions parameterized by elliptic curve periods.

Findings

Explicit solutions to Schlesinger systems constructed.

New families of Painlevé VI solutions expressed via elliptic integrals.

Parameter-dependent solutions for various integer values of n.

Abstract

We construct algebro-geometric upper triangular solutions of rank two Schlesinger systems. Using these solutions we derive two families of solutions to the sixth Painlev\'e equation with parameters $({1}/{8}, -{1}/{8}, {1}/{8}, {3}/{8})$ expressed in simple forms using periods of differentials on elliptic curves. Similarly for every integer $n$ different from $0$ and $-1$ we obtain one family of solutions to the sixth Painlev\'e equation with parameters $(\frac{9n^2+12n+4}{8}, -\frac{n^2}{8}, \frac{n^2}{8}, \frac{4-n^2}{8})$.