  In this note we analyze smooth solutions of a $p$-system of the
\textit{mixed} type. Motivating example for this is a 2-components reduction of
the Benney moments chain which appears to be connected to theory of integrable
systems. We don't assume a-priory that the solutions in question are in the
Hyperbolic region. Our main result states that the only smooth solutions of the
system which are periodic in $x$ are necessarily constants. As for initial
value problem we prove that if the initial data is strictly hyperbolic and
periodic in $x$ then the solution can not extend to $[t_0;+\infty)$ and shocks
are necessarily created.
