This paper investigates the spatio-temporal symmetries of periodic trajectories in dynamical systems with S-N and S-N x Z(2) symmetry. It turns out that trajectories in S-N-equivariant systems cannot exhibit spatio-temporal symmetries beyond the trivial symmetry of all periodic orbits. More complex symmetries in the trajectories require additional constraints on the dynamics. The possibilities offered by S-N x Z(2) symmetric systems are considered and a specific S-3 x Z(2)-equivariant system is investigated numerically
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