Note on algebro-geometric solutions to triangular Schlesinger systems

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 Painlevé 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 Painlevé equation with parameters (Figure presented.).