  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.
