Characterization of affine G_m-surfaces of hyperbolic type

In this note we prove that if S is an affine non-toric G_m-surface of hyperbolic type that admits a G_a-action and X is an affine irreducible variety such that Aut(X) is isomorphic to Aut(S) as an abstract group, then X is a Gm-surface of hyperbolic type. Further, we show that a smooth Danielewski surface D_p = {xy=p(z)} ⊂ A^3, where p has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.