Picard and Chazy solutions to the Painleve VI equation

We study the solutions of a particular family of Painleve VI equations with parameters beta = gamma = 0, delta = 1/2 and 2 alpha = (2 mu - 1)(2), for 2 mu epsilon Z. We show that in the case of half-integer mu, 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, infinity and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVImu equation for any mu such that 2 mu is an element of Z. As an application, we classify all the algebraic solutions. For mu half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For mu integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions.