  Certain features associated with the symmetry reduction of the vacuum
Einstein equations by two commuting, space-like Killing vector fields are
studied. In particular, the discussion encompasses the equations for the Gowdy
$T^3$ cosmology and cylindrical gravitational waves. We first point out a
relation between the $SL(2,R)$ (or SO(3)) $\sigma$ and principal chiral models,
and then show that the reduced Einstein equations can be obtained from a
dimensional reduction of the standard SL(2,R) sigma-model in three dimensions.
The reduced equations can also be derived from the action of a `generalized'
two dimensional SL(2,R) sigma-model with a time dependent constraint. We give a
Hamiltonian formulation of this action, and show that the Hamiltonian evolution
equations for certain phase space variables are those of a certain
generalization of the principal chiral model. Using these Hamiltonian
equations, we give a prescription for obtaining an infinite set of constants of
motion explicitly as functionals of the space-time metric variables.
