Abstract
In this note we extend the result from [14] and prove that if S is an affine non-toric Gm-surface of hyperbolic type that admits a Ga-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 Dp={xy=p(z)}⊂A3, where p has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.
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