  The aim of this work is twofold. From a mathematical point of view, we show
the existence of a hyperbolic system of equations that is not symmetrizable in
the sense of Friedrichs. Such system appears in the theory of compressible
fluid dynamics with Cattaneo-type extensions for the heat flux. In contrast,
the linearizations of such system around constant equilibrium solutions have
Friedrichs symmetrizers. Then, from a physical perspective, we aim to
understand the relaxation term appearing in this system. By noticing the
violation of the Kawashima-Shizuta condition, locally and smoothly, with
respect to the Fourier frequencies, we construct persistent waves, i.e.,
solutions preserving the $L^{2}$ norm for all times that are not dissipated by
the relaxation terms.
