Constants of motion for vacuum general relativity

The 3+1 Hamiltonian Einstein equations, reduced by imposing two commuting spacelike Killing vector fields, may be written as the equations of the SL(2,$R$) principal chiral model with certain “source” terms. Using this formulation, we give a procedure for generating an infinite number of nonlocal constants of motion for this sector of the Einstein equations. The constants of motion arise as explicit functionals on the phase space of Einstein gravity, and are labeled by sl(2,$R$) indices.