  We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum
spacetime with a space-like translational Killing field in the asymptotically
flat case.. The presence of a space-like translational Killing field allows for
a reduction of the 3 + 1 dimensional problem to a 2+1 dimensional one. The aim
of this paper is to go further in the asymptotic expansion of the solutions
than in [14]. In particular the expansion we construct involves quantities
which are the 2-dimensional equivalent of the global charges.
