Darboux-Egorov system, bi-flat $F$-manifolds and Painlev\'e VI

This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e coefficients $H_i$ involved in the construction are still homogeneous functions of a certain degree $d_i$ but we consider the general case $d_i\ne d_j$. As a consequence the rotation coefficients $\beta_{ij}$ are homogeneous functions of degree $d_i-d_j-1$. It turns out that any semisimple bi-flat $F$ manifold satisfying a natural additional assumption can be obtained in this way. Finally we show that three dimensional semisimple bi-flat $F$-manifolds are parametrized by solutions of the full family of Painlev\'e VI.