Confluence of hypergeometric functions and integrable hydrodynamic-type systems

We construct a new class of integrable hydrodynamic-type systems governing the dynamics of the critical points of confluent Lauricella-type functions defined on finite-dimensional Grassmannian Gr(2, n ), i.e., on the set of 2× n matrices of rank two. These confluent functions satisfy certain degenerate Euler–Poisson–Darboux equations. We show that in the general case, a hydrodynamic-type system associated with the confluent Lauricella function is an integrable and nondiagonalizable quasilinear system of a Jordan matrix form. We consider the cases of the Grassmannians Gr(2, 5) for two-component systems and Gr(2, 6) for three-component systems in detail.