INTEGRALS OF MOTION IN THE 2-KILLING-VECTOR REDUCTION OF GENERAL-RELATIVITY

We apply the inverse scattering method to the midi-superspace models t hat are characterized by a two-parameter abelian group of motions with two spacelike Killing vectors. We present a formulation that simplifi es the construction of the soliton solutions of Belinskii and Zakharov . Furthermore, it enables us to obtain the zero-curvature formulation for these models. Using this, and imposing periodic boundary condition s corresponding to the Gowdy models when the spatial topology is a thr ee-torus T3, we show that the equation of motion for the monodromy mat rix is an evolution equation of Heisenberg type. Consequently, the eig envalues of the monodromy matrix are the generating functionals for th e integrals of motion. Furthermore, we utilise a suitable formulation of the transition matrix to obtain explicit expressions for the integr als of motion. This involves recursion relations which arise in solvin g an equation of Riccati type. In the case when the two Killing vector s are hypersurface orthogonal the integrals of motion have a particula rly simple form.