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We analyze here the minimal conditions for directional motion (net flow in phase space) of a molecular motor placed on a mirror-symmetric environment and driven by a center-symmetric and time-periodic force field. The complete characterization of the deterministic limit of the dissipative dynamics of several realizations of this minimal model reveals a complex structure in the phase diagram in parameter space, with intertwined regions of pinning (closed orbits) and directional motion. This demonstrates that the mirror symmetry breaking, which is needed for directional motion to occur, can operate through an internal degree of freedom coupled to the translational one.