2006.386: Picard and Chazy Solutions to the Painlevé VI Equation

2006.386: Marta Mazzocco (2001) Picard and Chazy Solutions to the Painlevé VI Equation. Mathematische Annalen, 321 (1). pp. 157-195. ISSN 0025-5831

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DOI: 10.1007/s002080100226

Abstract

We study the solutions of a particular family of Painlevé 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 PVI_μ 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.

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