On periodic solutions for a reduction of benney chain

We study periodic solutions for a quasi-linear system, which naturally arises in search of integrable Hamiltonian systems of the form H = p²/2 + u(q, t). Our main result classifies completely periodic solutions for such a 3 by 3 system. We prove that the only periodic solutions have the form of traveling waves so, in particular, the potential u is a function of a linear combination of t and q. This result implies that the there are no nontrivial cases of the existence of a fourth power integral of motion for H: if it exists, then it is equal necessarily to the square of a quadratic integral. Our main observation for the quasi-linear system is the genuine non-linearity of the maximal and minimal eigenvalues in the sense of Lax. We use this observation in the hyperbolic region, while the "elliptic" region is treated using the maximum principle.