  In this paper we show that all supergravity billiards corresponding to
sigma-models on any U/H non compact-symmetric space and obtained by
compactifying supergravity to D=3 are fully integrable. The key point in
establishing the integration algorithm is provided by an upper triangular
embedding of the solvable Lie algebra associated with U/H into SL(N,R) which
always exists. In this context we establish a remarkable relation between the
arrow of time and the properties of the Weyl group. The asymptotic states of
the developing Universe are in one-to-one correspondence with the elements of
the Weyl group which is a property of the Tits Satake universality classes and
not of their single representatives. Furthermore the Weyl group admits a
natural ordering in terms of L(T), the number of reflections with respect to
the simple roots and the direction of time flows is always towards increasing
L(T), which plays the unexpected role of an entropy.
