  In this paper we pursue further a programme initiated in a previous work and
aimed at the construction, classification and property investigation of time
dependent solutions of supergravity (superstring backgrounds) through a
systematic exploitation of U-duality hidden symmetries. This is done by first
reducing to D=3 where the bosonic part of the theory becomes a sigma model on
E_{8(8)}/SO(16), solving the equations through an algorithm that produces
general integrals for any chosen regular subalgebra G_r of E_{8(8)} and then
oxiding back to D=10. Different oxidations and hence different physical
interpretations of the same solutions are associated with different embeddings
of G_r. We show how such embeddings constitute orbits under the Weyl group and
we study the orbit space. This is relevant to associate candidate superstring
cosmological backgrounds to space Dp-brane configurations that admit
microscopic descriptions. In particular in this paper we show that there is
just one Weyl orbit of A_r subalgebras for r < 6$. The orbit of the previously
found A_2 solutions, together with space--brane representatives contains a pure
metric representative that corresponds to homogeneous Bianchi type 2A
cosmologies in D=4 based on the Heisenberg algebra. As a byproduct of our
methods we obtain new exact solutions for such cosmologies with and without
matter. We present a thorough investigation of their properties.
