Phase spaces and symmetries of Vaidya superspace

We investigate the classical symmetries of the dynamics of the null-dust spherically symmetric Vaidya spacetime. Einstein's equations for this model can be obtained as equations of motion of a two-dimensional field theory. We discuss the transformations leaving invariant such equations of motion. These are given by two distinct sets, the residual diffeomorphisms coming from general relativity and the generalisation of the Schr\"odinger symmetry, recently found for the static Schwarzschild black holes. Surprisingly, these two sets represent the symmetries of two different action functionals, leading to the same equations of motion, but with different phase spaces.