We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum space-time 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.