Laplace transformations of hydrodynamic-type systems in Riemann invariants

The conserved densities of hydrodynamic-type systems in Riemann invariants satisfy a system of linear second-order partial differential equations. For linear systems of this type, Darboux introduced Laplace transformations, which generalize the classical transformations of a second-order scalar equation. It is demonstrated that the Laplace transformations can be pulled back to transformations of the corresponding hydrodynamic-type systems. We discuss finite families of hydrodynamic-type systems that are closed under the entire set of Laplace transformations. For $3\times3$ systems in Riemann invariants, a complete description of closed quadruples is proposed. These quadruples appear to be related to a special quadratic reduction of the $(2+1)$-dimensional 3-wave system.