Mathematische Annalen

, Volume 321, Issue 1, pp 157–195 | Cite as

Picard and Chazy solutions to the Painlevé VI equation

  • Marta Mazzocco
Original article


We study the solutions of a particular family of Painlevé VI equations with parameters \(\beta=\gamma=0, \delta=\frac{1}{2}\) and \(2\alpha=(2\mu-1)^2\), for \(2\mu\in{\mathbb Z}\). 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 \(0,1,\infty\) and their nonlinear monodromy. We 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, 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.


Asymptotic Behaviour Singular Point Explicit Formula Classical Solution Analytic Continuation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Marta Mazzocco
    • 1
  1. 1.Mathematical Insitute, 24–29 St. Giles, Oxford OX1 3LB, UK GB

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