  We investigate the phase diagram of a two-dimensional driven-dissipative
system of polaritons coupled to the excitonic reservoir. We find that two
critical points exists. The first corresponds to the quasi-condensation and the
second to a first-order phase transition from the non-uniform state with
spatially modulated density to a uniform state. The latter is related to the
modulational instability of a homogeneous state due to the repulsive
interactions with the non-condensed reservoir. The first-order character of the
transition is evidenced by a discontinuity in the density and the correlation
length as well as the phase coexistence and metastability. Moreover, we show
that a signature of a Berezinskii-Kosterlitz-Thouless-like transition can be
observed in the non-uniform phase.
