  We develop a systematic algorithm to construct, classify and study exact
solutions of type II A/B supergravity which are time--dependent and homogeneous
and hence represent candidate cosmological backgrounds. Using the formalism of
solvable Lie algebras to represent the geometry of non--compact coset manifolds
U/H we are able to reduce the supergravity field equations to the geodesic
equations in U/H and rephrase these latter in a completely algebraic setup by
means of the so called Nomizu operator representation of covariant derivatives
in solvable group manifolds. In this way a systematic method of integration of
supergravity equations is provided. We show how the possible D=3 solutions are
classified by non--compact subalgebras G of E8(8) and their ten--dimensional
physical interpretation (oxidation) depends on the classification of the
different embeddings of G inside E8(8). We give some preliminary examples of
explicit solutions based on the simplest choice G=A2. We also show how, upon
oxidation, these solutions provide a smooth and exact realization of the
bouncing phenomenon on Weyl chamber walls envisaged by the cosmological
billiards of Damour et al. We also show how this physical phenomenon is
triggered by the presence of euclidean $D$--branes possibly interpretable at
the microscopic level as S--branes. We outline how our analysis could be
extended to a wider setup where, by further reducing to $D=2,1$, more general
backgrounds could be constructed applying our method to the infinite algebras
E9 and E10.
