Characterization of affine Gm-surfaces of hyperbolic type

In this note we extend the result from [14] and 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 $\mathrm{Aut}\left(X\right)$ is isomorphic to $\mathrm{Aut}\left(S\right)$ as an abstract group, then X is a $G_m$-surface of hyperbolic type. Further, we show that a smooth Danielewski surface $D_p=\{xy=p(z\left)\right\}\subset A^3$, where p has no multiple roots, is determined by its automorphism group seen as an ind-group in the category of affine irreducible varieties.