Picard and Chazy solutions to the Painleve' VI equation

Abstract

I study the solutions of a particular family of Painlevé VI equations with the parameters $β=γ=0, δ=1/2$ and $2α=(2μ-1)^2$, for $2μ\in\interi$. I show that the case of half-integer $μ$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$μ$ equation for any $μ$ such that $2μ\in\interi$. As an application, I classify all the algebraic solutions. For $μ$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $μ$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.