  We prove that a normal affine surface $V$ over $\bf C$ admits an effective
action of a maximal torus ${\bf T}={\bf C}^{*n}$ ($n\le 2$) such that any other
effective ${\bf C}^*$-action is conjugate to a subtorus of $\bf T$ in Aut
$(V)$, in the following particular cases: (a) the Makar-Limanov invariant
ML$(V)$ is nontrivial, (b) $V$ is a toric surface, (c) $V={\bf P}^1\times {\bf
P}^1\backslash \Delta$, where $\Delta$ is the diagonal, and (d) $V={\bf
P}^2\backslash Q$, 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) and (d) is a result of
Danilov-Gizatullin and Doebeli.
