  The existence of a dissipative flux vector is known to be compatible with
reversible processes, provided a timelike conformal Killing vector (CKV)
$\chi^\alpha=\frac{V^\alpha}{T}$ (where $V^\alpha$ and $T$ denote the
four-velocity and temperature respectively) is admitted by the space-time. Here
we show that if a constitutive transport equation, either within the context of
standard irreversible thermodynamics or the causal Israel--Stewart theory, is
adopted, then such a compatibility also requires vanishing dissipative fluxes.
Therefore, in this later case the vanishing of entropy production generated by
the existence of such CKV is not actually associated to an imperfect fluid, but
to a non-dissipative one. We discuss also about Landau damping.
