For the analysis of the long-time behaviour of mechanical systems with symmetry it is crucial to understand the bifurcations of REs and RPOs as internal parameters such as energy and other conserved quantities are varied. Whereas the theory of generic symmetry breaking bifurcations of such invariant sets is well-developed for general (non-Hamiltonian) systems, there are many fewer results on the corresponding theory for symmetric mechanical systems. This is due to the various conservation laws of mechanical systems with symmetry which change the generic behaviour of a dynamical system drastically and therefore have to be taken into account. So far a systematic numerical bifurcation analysis only exists for equilibria and periodic orbits of non-symmetric systems. We aim at the parallel development of theoretical and numerical methods for symmetry breaking bifurcations of simple invariant sets of symmetric mechanical systems using the code SYMPERCON. The results will be applied to various examples of mechanical systems from the areas mentioned above.