Algebraic Solutions to the Painlevé-VI Equation and Reflection Groups

We study the global analytic properties of the solutions of a particular family of Painleve VI equations with the parameters (3 = I = 0, J = ~ and 2a = (2μ - 1 )2 , with μ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. The finite orbits of this action correspond to the algebraic solutions of our Painleve VI equation. For 2μ f/:. "ll., the auxiliary reflection group is always irreducible. For μ integer, the auxiliary reflection group is either irreducible or trivial (i.e. it contains only the identity) and for μ half-integer it always reduces to an irreducible reflection group in the two dimensional space. We classify all the finite orbits of the action of the braid group on the irreducible reflection groups in the three-dimensional and in the two-dimensional space. It turns out that for all these orbits μ is not integer. This result is used to classify all the algebraic solutions to our Painleve VI equation with μ f/:. "ll.. For 2μ f/:. "ll., they are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space, for half-integerμ they are in one-to-one correspondence with the regular polygons or star-polygons in the plane. For integerμ, the only algebraic solutions all belong to a one-parameter family of rational solutions and correspond to the trivial auxiliary reflection group. Moreover, we show that the case of half-integerμ is integrable, and that its solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. We give explicit formulae for them, completely describe the asymptotic behaviour around the critical points O, 1, oo and the non linear monodromy.