The fifty-two icosahedral solutions to Painlevé VI

Abstract

The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F₄ Weyl group action and many properties of the solutions will be given.

There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known P_VI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev.

Of the remaining 28 solutions we will find 20 explicitly using the method of [P. P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. 90 (2005), no. 3, 167–208.] (via Jimbo's asymptotic formula). Amongst those constructed there is one solution that is “generic” in that its parameters lie on none of the affine F₄ hyperplanes, one that is equivalent to the Dubrovin-Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.