  We consider 1 spacelike Killing vector field reductions of 4-d vacuum general
relativity. We restrict attention to cases in which the manifold of orbits of
the Killing field is $R^{3}$. The reduced Einstein equations are equivalent to
those for Lorentzian 3-d gravity coupled to an SO(2,1) nonlinear sigma model on
this manifold. We examine the theory in terms of a Hamiltonian formulation
obtained via a 2+1 split of the 3-d manifold. We restrict attention to
geometries which are asymptotically flat in a 2-d sense defined recently. We
attempt to pass to a reduced Hamiltonian description in terms of the true
degrees of freedom of the theory via gauge fixing conditions of 2-d conformal
flatness and maximal slicing. We explicitly solve the diffeomorphism
constraints and relate the Hamiltonian constraint to the prescribed negative
curvature equation in $R^2$ studied by mathematicians. We partially address
issues of existence and/or uniqueness of solutions to the various elliptic
partial differential equations encountered.
