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), for 2μ ∈ Z . 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,∞ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVIμ equation for any μ such that 2μ ∈ Z . 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.