In this paper we complete the classification of effective C ∗-actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ ↦ → λ −1 of C ∗. If a smooth affine surface V admits more than one C ∗-action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [FKZ3] we gave a sufficient condition, in terms of the Dolgachev-Pinkham-Demazure (or DPD) presentation, for the uniqueness of a 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 C ∗-actions depending on one or two parameters. We give an explicit description of all such surfaces and their 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-conjugat
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