  We investigate the occurrence of a phase transition, characterized by the
spontaneous breaking of a discrete symmetry, in a driven-dissipative
Bose-Hubbard lattice in presence of two-photon coherent driving. The driving
term does not lift the original $U(1)$ symmetry completely and a discrete
$\mathbb{Z}_2$ symmetry is left. When driving the bottom of the Bose-Hubbard
band, a mean-field analysis of the steady state reveals a second-order
transition from a symmetric phase to a quasi-coherent state with a finite
expectation value of the Bose field. For larger driving frequency, the phase
diagram shows a third region, where both phases are stable and the transition
becomes of first order.

Quantum Physics