  We consider the existence, stability and dynamics of the nodeless state and
fundamental nonlinear excitations, such as vortices, for a
quasi-two-dimensional polariton condensate in the presence of pumping and
nonlinear damping. We find a series of interesting features that can be
directly contrastedto the case of the typically energy-conserving ultracold
alkali-atom Bose-Einstein condensates (BECs). For sizeable parameter ranges, in
line with earlier findings, the nodeless state becomes unstable towards the
formation of {\em stable} nonlinear single or multi-vortex excitations. The
potential instability of the single vortex is also examined and is found to
possess similar characteristics to those of the nodeless cloud. We also report
that, contrary to what is known, e.g., for the atomic BEC case, {\it stable}
stationary gray rings (that can be thought of as radial forms of a Nozaki-Bekki
hole) can be found for polariton condensates in suitable parametric regimes. In
other regimes, however, these may also suffer symmetry breaking instabilities.
The dynamical, pattern-forming implications of the above instabilities are
explored through direct numerical simulations and, in turn, give rise to
waveforms with triangular or quadrupolar symmetry.
