Degenerative dispersion laws, motion invariants and kinetic equations

Let us consider the following problem. Imagine a homogeneous nonlinear medium in which only one type of wave with dispersion law co(k) may propagate. Let the nonlinearity of the medium be quadratic while the equation co(k~ + k~) = co(k0 + co(k2) (1) defines a nonzero manifold F whose codimension in kl, k2 space is unity. Equation (1) means that dispersion law co(k) allows decay processes. The nonlinearity being weak, waves in such a system may be described statistically by introducing the average occupation numbers nk of the state with momentum k. The time evolution of nk is governed by the kinetic equation