Smooth affine surfaces with non-unique $\mathbb {C}^*$-actions

In this paper we complete the classification of effective $\mathbb {C}^*$-actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion $\lambda \mapsto \lambda ^{-1}$ of $\mathbb {C}^*$. If a smooth affine surface $V$ admits more than one $\mathbb {C}^*$-action, then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [Transformation Groups 13:2, 2008, pp. 305–354] we gave a sufficient condition, in terms of the Dolgachev-Pinkham-Demazure (or DPD) presentation, for the uniqueness of a $\mathbb {C}^*$-action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface $V$ which is neither toric nor Danilov-Gizatullin, then $V$ admits a continuous family of pairwise non-conjugated $\mathbb {C}^*$-actions depending on one or two parameters. We give an explicit description of all such surfaces and their $\mathbb {C}^*$-actions in terms of DPD presentations. We also show that for every $k>0$ one can find a Danilov-Gizatullin surface $V(n)$ of index $n=n(k)$ with a family of pairwise non-conjugate $\mathbb {C}_+$-actions depending on $k$ parameters.