We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 1 2 and 2α = (2µ−1) 2 with arbitrary µ, 2µ ∈ Z. 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. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space.

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) 2 , for 2µ ∈ Z 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 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.SISSA preprint no. 89/98/FM,

The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.

In this paper, we classify all values of the parameters α, β, γ and δ of the Painlevé VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlevé VI equation.

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