An exact asymptotically flat four-parameter solution of the Einstein-M
axwell equations representing the exterior field of two identical Kerr
-Newman sources with angular momenta oppositely orientated is construc
ted in a simple analytical form. We show that there are no equilibrium
states in such compact systems except for the case when the masses of
the sources are equal to their charges. The condition for the existen
ce of two horizons in the 'antisymmetric' binary system turns out to b
e more stringent than in the case of a single Kerr-Newman black hole.