  General Relativity reduced to two dimensions possesses a large group of
symmetries that exchange classical solutions. The associated Lie algebra is
known to contain the affine Kac-Moody algebra $A_1^{(1)}$ and half of a real
Witt algebra. In this paper we exhibit the full symmetry under the semi-direct
product of $\Lie{A_1^{(1)}}$ by the Witt algebra $\Lie{\Wir}$. Furthermore we
exhibit the corresponding hidden gauge symmetries. We show that the theory can
be understood in terms of an infinite dimensional potential space involving all
degrees of freedom: the dilaton as well as matter and gravitation. In the
dilaton sector the linear system that extends the previously known Lax pair has
the form of a twisted self-duality constraint that is the analog of the
self-duality constraint arising in extended supergravities in higher spacetime
dimensions. Our results furnish a group theoretical explanation for the
simultaneous occurrence of two spectral parameters, a constant one ($=y$) and a
variable one ($=t$). They hold for all $2d$ non-linear $\sigma$-models that are
obtained by dimensional reduction of $G/H$ models in three dimensions coupled
to pure gravity. In that case the Lie algebra is $\Lie{\Wir \semi G^{(1)}}$;
this symmetry acts on a set of off shell fields (in a fixed gauge) and
preserves the equations of motion.

High Energy Physics - Theory;General Relativity and Quantum Cosmology;Exactly Solvable and Integrable Systems;Exactly Solvable and Integrable Systems