  Condensation of exciton-polaritons has been recently observed in
one-dimensional photonic crystal waveguides, exploiting the interplay of
long-lived gap confined eigenmodes and negative mass polariton branches. Here
we focus on the theoretical emergence of a second emission threshold, in
addition to the one associated with condensation at zero-momentum, due to the
nonlinear polariton scattering from the condensate into finite momentum
eigenmodes. The physics of this spatially modulated condensate is related to a
spontaneous breaking of both phase and translational symmetries simultaneously,
bearing strong similarities with the highly sought supersolid phase in Helium
and ultracold atomic gases but with a novel mechanism typical of the
driven-dissipative scenario. We then propose clear-cut and unequivocal
experimental signatures that would allow to identify supersolidity phenomena in
polariton condensates
