Title | Picard and Chazy solutions to the Painlevé VI equation |

Publication Type | Journal Article |

Year of Publication | 2001 |

Authors | Mazzocco, M |

Journal | Math. Ann. 321 (2001) 157-195 |

Abstract | 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. |

URL | http://hdl.handle.net/1963/3118 |

DOI | 10.1007/PL00004500 |

## Picard and Chazy solutions to the Painlevé VI equation

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