01274nas a2200109   4500008004300000245006000043210006000103260001300163520093200176100002001108856003601128       2001                        en_Ud  00aPicard and Chazy solutions to the Painlevé VI equation0 aPicard and Chazy solutions to the Painlevé VI equation  bSpringer3 a<p>I study the solutions of a particular family of Painlevé VI equations with the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for $2\mu\in\mathbb{Z}$. I show that the case of half-integer $\mu$ 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$\mu$ equation for any $\mu$ such that $2\mu\in\mathbb{Z}$. As an application, I classify all the algebraic solutions. For $\mu$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $\mu$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.</p>1 aMazzocco, Marta  uhttp://hdl.handle.net/1963/3118