Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras

In this paper we show that the supergravity equations describing both cosmic billiards and a large class of black-holes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on the dual Borel algebra B_N of the simple Lie algebra A_{N-1}. As a by product we derive the explicit integration algorithm associated with all symmetric spaces U/H^{*} relevant to the description of time-like and space-like p-branes. The most important consequence of our approach is the explicit construction of a complete set of conserved involutive hamiltonians h_{\alpha} that are responsible for integrability and provide a new tool to classify flows and orbits. We believe that these will prove a very important new tool in the analysis of supergravity black holes and billiards.