On the uniqueness of C*-actions on affine surfaces

It is an open question whether every normal affine surface V over C admits an effective action of a maximal torus T = C*(n) (n <= 2) such that any other effective C*-action is conjugate to a subtorus of T in Aut(V). We prove that this holds indeed in the following cases: (a) the Makar-Limanov invariant ML(V) not equal C is nontrivial, (b) V is a toric surface, (c) V = P-1 x P-1\Delta, where Delta is the diagonal, and (d) V = P-2\Q, where Q is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth [Gut]) and (d) is a result of Danilov-Gizatullin [DG] and Doebeli [Do].