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Zbl 0999.34079
Mazzocco, Marta
Picard and Chazy solutions to the Painlevé VI equation. (English)
[J] Math. Ann. 321, No.1, 157-195 (2001). ISSN 0025-5831; ISSN 1432-1807/e

Here, the author studies the Painlevé equation $PVI_\mu$ with the parameters $\beta=\gamma=0$, $\delta=\frac 12$ and $2\alpha=(2\mu-1)^2$ for half-integer $\mu$: $$w''=\frac 12\left(\frac 1w+\frac{1}{w-1}+\frac{1}{w-z}\right){w'}^2-\left(\frac 1z +\frac{1}{z-1}+\frac{1}{w-z}\right)w'$$ $$+\frac{1}{2} \frac{w(w-1)(w-z)}{z^2(z-1)^2} \left[(2\mu-1)^2 + \frac{z(z-1)}{(w-z)^2}\right].$$ He shows that, for any half-integer $\mu$, all solutions to the $PVI_\mu$ equation can be computed in terms of known special functions. All solutions be divided into two types: (1) a two-parameter family of solutions found by Picard; (2) a new one-parameter family of classical solutions which be called Chazy solutions. The author gives explicit formulae for them and completely determines their asymptotic behaviour near the singular points 0, 1, $\infty$ and their nonlinear monodromy. He studies the structure of analytic continuation of the solutions to the $PVI_\mu$ equation for any half-integer $\mu$. For $\mu$ half-integer, the author shows that all algebraic functions are in one to one correspondence with regular polygons or star-polygons in the plane. For $\mu$ integer, he shows that all algebraic solutions belong to a one-parameter family of rational solutions.
[Chen Zong-xuan]

MSC 2000:: *34M55 Painlevé and other special equations

Keywords: Picard solutions; Chazy solutions; Painlevé IV equation; rational solutions

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