  The formulation of gravity and M-theories as very-extended Kac-Moody
invariant theories encompasses, for each very-extended algebra G+++, two
distinct actions invariant under the overextended Kac-Moody subalgebra G++. The
first carries a Euclidean signature and is the generalisation to G++ of the
E10-invariant action proposed in the context of M-theory and cosmological
billiards. The second action carries various Lorentzian signatures revealed
through various equivalent formulations related by Weyl transformations of
fields. It admits exact solutions, identical to those of the maximally oxidised
field theories and of their exotic counterparts, which describe intersecting
extremal branes smeared in all directions but one. The Weyl transformations of
G++ relates these solutions by conventional and exotic dualities. These exact
solutions, common to the Kac-Moody theories and to space-time covariant
theories, provide a laboratory for analysing the significance of the infinite
set of fields appearing in the Kac-Moody formulations.
