[en] We discuss the nature of symmetry breaking and the associated collective excitations for a system of bosons coupled to the electromagnetic field of two optical cavities. For the specific configuration realized in a recent experiment at ETH [1, 2], we show that, in absence of direct intercavity scattering and for parameters chosen such that the atoms couple symmetrically to both cavities, the system possesses an approximate U(1) symmetry which holds asymptotically for vanishing cavity field intensity. It corresponds to the invariance with respect to redistributing the total intensity $I=I_1+I_2$ between the two cavities. The spontaneous breaking of this symmetry gives rise to a broken continuous translation-invariance for the atoms, creating a supersolid-like order in the presence of a Bose–Einstein condensate. In particular, we show that atom-mediated scattering between the two cavities, which favors the state with equal light intensities $I_1=I_2$ and reduces the symmetry to ${\mathbf{Z}}_2\otimes <!--\; ???\; -->{\mathbf{Z}}_2$, gives rise to a finite value $\sim <!--\; ???\; -->\sqrt{I}$ of the effective Goldstone mass. For strong atom driving, this low energy mode is clearly separated from an effective Higgs excitation associated with changes of the total intensity I. In addition, we compute the spectral distribution of the cavity light field and show that both the Higgs and Goldstone mode acquire a finite lifetime due to Landau damping at non-zero temperature. (paper)