Integrability of Supergravity Billiards and the generalized Toda lattice   equation
  We prove that the field equations of supergravity for purely time-dependent backgrounds, which reduce to those of a one--dimensional sigma model, admit a Lax pair representation and are fully integrable. In the case where the effective sigma model is on a maximally split non--compact coset U/H (maximal supergravity or subsectors of lower supersymmetry supergravities) we are also able to construct a completely explicit analytic integration algorithm, adapting a method introduced by Kodama et al in a recent paper. The properties of the general integral are particularly suggestive. Initial data are represented by a pair C_0, h_0 where C_0 is in the CSA of the Lie algebra of U and h_0 in H/W is in the compact subgroup H modded by the Weyl group of U. At asymptotically early and asymptotically late times the Lax operator is always in the Cartan subalgebra and due to the iso-spectral property the two limits differ only by the action of some element of the Weyl group. Hence the entire cosmic evolution can be seen as a billiard scattering with quantized angles defined by the Weyl group. The solution algorithm realizes a map from H}/W into W. 
Pietro Fr\'e and Alexander Sorin