Chazy's second-degree Painlevé equations

Abstract

We examine two sets of second-degree Painlevé equations derived by Chazy in 1909, denoted by systems (II) and (III). The last member of each set is a second-degree version of the Painlevé-VI equation, and there are no other second-order second-degree Painlevé equations in the polynomial class with this property. We map the last member of system (II) into the Fokas–Yortsos equation and demonstrate how both Schlesinger and Okamoto transformations for Painlevé-VI can be read off the Chazy equation. The 24 fundamental Schlesinger transformations were known to Garnier in 1943 while the 64 Okamoto transformations date from 1987. In an appendix, we gather together the solutions of the five members of system (II). System (III) is better known, being equivalent to Jimbo and Miwa's equations for the logarithmic derivatives of the tau functions of the six Painlevé transcendents. The last member, known to Painlevé in 1906, was written in a manifestly symmetric form by Jimbo and Miwa, suggesting many induced symmetries for Painlevé-VI. In particular, Schlesinger and Okamoto transformations for Painlevé-VI can be read off immediately.