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 $\mathrm{Gr}(2,n)$, i. e., on the set of $2\times n$ 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 $\mathrm{Gr}(2,5)$ for two-component systems and $\mathrm{Gr}(2,6)$ for three-component systems in detail.