  Explicit symmetry breaking occurs when a dynamical system having a certain
symmetry group is perturbed in a way that the perturbation preserves only some
symmetries of the original system. We give a geometric approach to study this
phenomenon in the setting of equivariant Hamiltonian systems. A lower bound for
the number of orbits of equilibria and orbits of relative equilibria which
persist after a small perturbation is given. This bound is given in terms of
the equivariant Lyusternik-Schnirelmann category of the group orbit.
