  The Einstein equations for spacetimes with two commuting spacelike Killing
field symmetries are studied from a Hamiltonian point of view. The complexified
Ashtekar canonical variables are used, and the symmetry reduction is performed
directly in the Hamiltonian theory. The reduced system corresponds to the field
equations of the SL(2,R) chiral model with additional constraints.
  On the classical phase space, a method of obtaining an infinite number of
constants of the motion, or observables, is given. The procedure involves
writing the Hamiltonian evolution equations as a single `zero curvature'
equation, and then employing techniques used in the study of two dimensional
integrable models. Two infinite sets of observables are obtained explicitly as
functionals of the phase space variables. One set carries sl(2,R) Lie algebra
indices and forms an infinite dimensional Poisson algebra, while the other is
formed from traces of SL(2,R) holonomies that commute with one another. The
restriction of the (complex) observables to the Euclidean and Lorentzian
sectors is discussed.
  It is also shown that the sl(2,R) observables can be associated with a
solution generating technique which is linked to that given by Geroch.
