Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II

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 [Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case, available at: arXiv:1302.1473]. In particular the expansion we construct involves quantities which are the 2-dimensional equivalent of the global charges.