Characterization of affine surfaces with a torus action by their automorphism groups

In this paper we prove that if two normal affine surfaces $S$ and $S'$ have isomorphic automophism groups, then every connected algebraic group acting regularly faithfully on acts also $S'$. Moreover, is non-toric, show the dynamical type of a 1-torus action preserved in presence an additive action. We complex toric are determined by abstract structure their regular automorphism groups category using properties Cremona group. As generalization to arbitrary dimensions, varieties, with exception torus, uniquely varieties seen as ind-groups.