The vacuum Einstein equations for spacetimes with two commuting spacel
ike Killing field symmetries are studied using the Ashtekar variables.
The case of compact spacelike hypersurfaces which are three-tori is c
onsidered, and the determinant of the Killing two-torus metric is chos
en as the time gauge. The Hamiltonian evolution equations in this gaug
e may be rewritten as those of a modified SL(2) principal chiral model
with a time-dependent ''coupling constant,'' or equivalently, with ti
me-dependent SL(2) structure constants. The evolution equations have a
generalized zero-curvature formulation. Using this form, the explicit
time dependence of an infinite number of spatial-diffeomorphism-invar
iant phase-space functionals is extracted, and it is shown that these
are observables in the sense that they Poisson commute with the reduce
d Hamiltonian. An infinite set of observables that have SL(2) indices
is also found. This determination of the explicit time dependence of a
n infinite set of spatial-diffeomorphism-invariant observables amounts
to the solutions of the Hamiltonian Einstein equations for these obse
rvables.