Conformal internal symmetry of 2d σ-models coupled to gravity and a dilaton

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(W). 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 σ-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(W⋉G); this symmetry acts on a set of off shell fields (in a fixed gauge) and preserves the equations of motion.