Infinite-dimensional Hamiltonian systems with symmetry-breaking perturbations are considered. We assume that the unperturbed system has an equilibrium and symmetries. When they are perturbed, some of the symmetries are broken but the rest persist. In this situation the equilibrium of unperturbed system may persist but its stability may change. In this talk, we briefly review some results for persistence and stability obtained by Kapitula [cf. Physica D, 156 (2001)] and give new bifurcation results. Some numerical calculations for standing waves of a nonlinear Schr\"odinger equation are also presented as examples.