  We present exact results for the steady-state density matrix of a general
class of driven-dissipative systems consisting of a nonlinear Kerr resonator in
the presence of both coherent (one-photon) and parametric (two-photon) driving
and dissipation. Thanks to the analytical solution, obtained via the complex
P-representation formalism, we are able to explore any regime, including photon
blockade, multiphoton resonant effects, and a mesoscopic regime with large
photon density and quantum correlations. We show how the interplay between one-
and two-photon driving provides a way to control the multimodality of the
Wigner function in regimes where the semiclassical theory exhibits
multistability. We also study the emergence of dissipative phase transitions in
the thermodynamic limit of large photon numbers.
