  We search for smooth periodic solutions for the system of quasi-linear PDEs
known as the Lax dispersionless reduction of the Benney moments chain. It is
naturally related to the existence of a polynomial in momenta integral for a
Classical Hamiltonian system with 1,5 degrees of freedom. For the solution in
question it is not known a-priori if the system is elliptic or hyperbolic or of
mixed type. We consider two possible regimes for the solution. The first is the
case of only one real eigenvalue, where we can completely classify the
solutions. The second case of strict Hyperbolicity is really a challenge. We
find a remarkable 2 by 2 reduction which is strictly Hyperbolic but violates
the condition of genuine non-linearity.
