We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum space-time with a space-like translational Killing field. The presence of a space-like transla-tional Killing field allows for a reduction of the 3 + 1 dimensional problem to a 2 + 1 dimensional one. Vacuum Einstein equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history (see [3], [1], [4]). In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on R 2. The main difficulty in that case is due to the delicate inversion of the Laplacian on R 2. In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.