  We discuss dissipative chaos showing symmetries in the phase space and
nonclassical statistics for a parametrically driven nonlinear Kerr resonator
(PDNR). In this system an oscillatory mode is created in the process of
degenerate down-conversion of photons under interaction with a train of
external Gaussian pulses. For chaotic regime we demonstrate, that the
Poincar\'e section showing a strange attractor, as well as the resonator mode
contour plots of the Wigner functions display two-fold symmetry in the phase
space. We show that quantum-to-classical correspondence is strongly violated
for some chaotic regimes of the PDNR. Considering the second-order correlation
function we show that the high-level of photons correlation leading to
squeezing in the regular regime strongly decreases if the system transits to
the chaotic regime. Thus, observation of the photon-number correlation allows
to extract information about the chaotic regime.
