Darboux–Egorov System, Bi-flat F-Manifolds and Painlevé VI

This is a generalization of the procedure presented in [3] to construct semisimple bi-flat F-manifolds (M,∇⁽¹⁾,∇⁽²⁾,∘,*,e,E) starting from homogeneous solutions of degree −1 of the Darboux–Egorov system. The Lamé coefficients Hi involved in the construction are still homogeneous functions of a certain degree di but we consider the general case di≠dj. As a consequence the rotation coefficients βij are homogeneous functions of degree di−dj−1. It turns out that any semisimple bi-flat F-manifold satisfying a natural additional assumption can be obtained in this way. Finally, building on the well-known relation between the three-wave system and Painlevé VI, we show that three-dimensional, semisimple, bi-flat F-manifolds can be constructed starting from generic solutions of Painlevé VI.