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Picard and Chazy solutions to the Painlevé VI equation. (English) Zbl 0999.34079

Here, the author studies the Painlevé equation PVI μ with the parameters β=γ=0, δ=1 2 and 2α=(2μ-1) 2 for half-integer μ:

w '' =1 21 w+1 w-1+1 w-zw ' 2 -1 z+1 z-1+1 w-zw '
+1 2w(w-1)(w-z) z 2 (z-1) 2 (2μ-1) 2 +z(z-1) (w-z) 2 ·

He shows that, for any half-integer μ, all solutions to the PVI μ 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 PVI μ equation for any half-integer μ. For μ 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 μ integer, he shows that all algebraic solutions belong to a one-parameter family of rational solutions.

Reviewer: Chen Zong-xuan
MSC:
34M55Painlevé and other special equations; classification, hierarchies