  The question of (non-)uniqueness of one-dimensional self-similar solutions to
the Riemann problem for hyperbolic systems of gas dynamics in sets of
multi-dimensional admissible weak solutions was addressed in recent years in
several papers culminating in [17] with the proof that the Riemann problem for
the isentropic Euler system with a power law pressure is ill-posed if the
one-dimensional self-similar solution contains a shock. Natural question then
arises whether the same holds also for a more involved system of equations, the
full Euler system. After the first step in this direction was made in [1],
where the ill-posedness was proved in the case of two shocks appearing in the
self-similar solution, we prove in this paper that the presence of just one
shock in the self-similar solution implies the same outcome, i.e. the existence
of infinitely many admissible weak solutions to the multi-dimensional problem.
