Additional integrals of the motion of classical Hamiltonian wave systems

It is shown that a classical Hamiltonian wave system that possesses at least one additional integral of the motion with quadratic principal part has an infinite number of such integrals in the cases of both nondegenerate and degenerate dispersion laws. Conditions under which in a space of dimension $d\geqslant 2$ a system with nondegenerate dispersion law is completely integrable and its Hamiltonian can be reduced to normal form are found. In the case of a degenerate dispersion law integrals are not sufficient for complete integrability.