  We study the momentum and Hamiltonian constraints of vacuum Einstein
equations, within the Bowen-York formalism, for two interacting black holes in
close separation, with anti-parallel spins and anti-parallel linear momenta. We
give an analytical solution using perturbation theory. We also compute the
location and the shape of the apparent horizon which generically depend on all
the parameters, angles and the separation between the black holes. Our solution
only works for distances far away from the black holes. To gain more insight
close to the black holes, one has to go to the higher orders in perturbation
theory, which is a rather cumbersome process. But the solution presented here
can be of some use for numerical computations as the latter should match our
result for the described problem.
