  We classify symmetry-protected and symmetry-breaking dynamical solutions for
nonlinear saturable bosonic systems that display a non-hermitian
charge-conjugation symmetry, as realized in a series of recent groundbreaking
experiments with lasers and exciton polaritons. In particular, we show that
these systems support stable symmetry-protected modes that mirror the concept
of zero-modes in topological quantum systems, as well as symmetry-protected
power-oscillations with no counterpart in the linear case. In analogy to
topological phases in linear systems, the number and nature of
symmetry-protected solutions can change. The spectral degeneracies signalling
phase transitions in linear counterparts extend to bifurcations in the
nonlinear context. As bifurcations relate to qualitative changes in the linear
stability against changes of the initial conditions, the symmetry-protected
solutions and phase transitions can also be characterized by topological
excitations, which set them apart from symmetry-breaking solutions. The
stipulated symmetry appears naturally when one introduces nonlinear gain or
loss into spectrally symmetric bosonic systems, as we illustrate for
one-dimensional topological laser arrays with saturable gain and
two-dimensional flat-band polariton condensates with density-dependent loss.
