Solutions to the wave equation for commuting flows of dispersionless PDEs

Motivated by the viewpoint of integrable systems, we study commuting flows of 2-component quasilinear equations, reducing to investigate the solutions of the wave equation with non-constant speed. In this paper, we apply the reduction procedure of differential constraints to obtain a complete set of solutions of such an equation for some fixed velocities a(2)(u,v). As a result, we present some examples of Hamiltonian integrable systems (as the shallow water equations) with relative symmetries, conserved quantities and solutions.