  In dissipative quantum systems, strong symmetries can lead to the existence
of conservation laws and multiple steady states. In this work we investigate a
strong symmetry for bosonic atoms coupled to an optical cavity, an
experimentally relevant system, generalizing the adiabatic elimination
techniques and using numerically exact matrix product state methods. We show
that for ideal bosons coupled to the cavity multiple steady states exist and in
each symmetry sector a dissipative phase transition occurs at a different
critical point. This implies that phases of very different nature can coexist.
We find that the introduction of a slight breaking of the strong symmetry by a
small interaction term leads to a direct transition from multiple steady states
to a unique steady state. We point out the phenomenon of dissipative freezing,
the breaking of the conservation law at the level of individual realizations in
the presence of the strong symmetry. For a small breaking of the strong
symmetry we see that the behavior of the individual trajectories still shows
some signs of this dissipative freezing before it fades out for a larger
symmetry breaking terms.
