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The fifty-two icosahedral solutions to Painlevé VI. (English) Zbl 1112.34072
Summary: The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group are classified up to equivalence under Okamoto’s affine ${F}_{4}$ Weyl group action and many properties of the solutions are 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 find 20 explicitly using the method in the paper of the author [Proc. Lond. Math. Soc., III. Ser. 90, 167–208 (2005; Zbl 1070.34123)] (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 ${F}_{4}$ 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.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies