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Certain features associated with the symmetry reduction of the vacuumEinstein equations by two commuting, space-like Killing vector fields arestudied. In particular, the discussion encompasses the equations for the Gowdy$T^3$ cosmology and cylindrical gravitational waves. We first point out arelation 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 adimensional 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 aHamiltonian formulation of this action, and show that the Hamiltonian evolutionequations for certain phase space variables are those of a certaingeneralization of the principal chiral model. Using these Hamiltonianequations, we give a prescription for obtaining an infinite set of constants ofmotion explicitly as functionals of the space-time metric variables.Comment: 17 pages, latex, reference added, a subtlety commented on; to appear  in Int. J. Mod. Phys.

SYMMETRY REDUCED EINSTEIN GRAVITY AND GENERALIZED σ AND CHIRAL MODELS