Picard and Chazy solutions to the Painlevé VI equation

We study the solutions of a particular family of Painlevë VI equations with parameters β = γ = 0, δ = 1/2 and 2α = (2μ-1)2, for 2μ ∈ ℤ. We show that in the case of half-integer μ, all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0, 1, ∞ and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVIμ equation for any μ such that 2μ ∈ ℤ. As an application, we classify all the algebraic solutions. For μ half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For μ integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions. © Springer-Verlag 2001.