Here, the author studies the Painlevé equation with the parameters , and for half-integer :
He shows that, for any half-integer , all solutions to the 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, and their nonlinear monodromy. He studies the structure of analytic continuation of the solutions to the equation for any half-integer . For 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 integer, he shows that all algebraic solutions belong to a one-parameter family of rational solutions.