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 $(\frac{9n^2+12n+4}{8},-\frac{n^2}{8},\frac{n^2}{8},\frac{4-n^2}{8})$.

12 pages