<p>In this paper, we find a class of nonstrictly hyperbolic systems of quasilinear equations with oscillatory solutions of the Cauchy problem, globally smooth in time in some open neighborhood of the zero stationary state. For such systems, the period of oscillation of solutions does not depend on the initial point of the Lagrangian trajectory. Also we discuss the possibility of constructing these systems in a physical context. We study nonrelativistic and relativistic equations of cold plasma from this point of view.</p>