*(English)*Zbl 0999.34079

Here, the author studies the Painlevé equation $PV{I}_{\mu}$ with the parameters $\beta =\gamma =0$, $\delta =\frac{1}{2}$ and $2\alpha ={(2\mu -1)}^{2}$ for half-integer $\mu $:

He shows that, for any half-integer $\mu $, all solutions to the $PV{I}_{\mu}$ 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 $PV{I}_{\mu}$ 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.

##### MSC:

34M55 | Painlevé and other special equations; classification, hierarchies |