  The knowledge of the properties of the different exact solutions modeling
binary systems, is a necessary step towards the classification of physically
suitable solutions and its corresponding limits of applicability. In the
present paper, we perform an analysis of the geodesics around two
counter--rotating Kerr--Newman black holes endowed with opposite electric
charges, which achieve equilibrium by means of a strut between their
constituents. We find that bounded and unbounded orbits are possible. However,
test particles may cross between the black holes only if their angular momentum
equals zero, otherwise, there exist a repulsive potential, which prohibits such
orbits. Two important aspects are pointed out for these trajectories: ({\it i})
the motion of photons is affected once crossing the strut; and ({\it ii})
massive particles exhibit oscillatory motion, as a first analog of the Sitnikov
problem in general relativity. The radius of the innermost stable circular
orbit as a function of the physical parameters of the black holes is also
investigated.
