A note on the R. Fuchs's problem for the Painlevé equations
arXiv:1204.0157
Abstract
In this article we consider a first-order completely integrable system of partial differential equations $\partial \Fi/partial x=A(x, t) \Fi, \partial \Fi/partial t=B(x, t) \Fi$ with $\Fi=(\fi_1, \fi_2)^τ$ where and are 2 by 2 holomorphic matrices functions. Under some assumptions we find a variable change by which the system $\partial \Fi/\partial x=A(x, t) \Fi$ is reduced to an equation independent on the variable . As an application we show that the R. Fuchs's conjecture for the Painlevé equations is true for some algebraic solutions.
16 pages