The isomorphism theorem for special surfaces

(This is a report on joint work with S. Kaliman and M. Zaidenberg)
In this talk we give a full classification of effective $\C^*$-actions on smooth affine
surfaces up to conjugation in the full automorphism group and up
to inversion $\lambda\mapsto \lambda^{-1}$ of $\C^*$. Our main result is:
Theorem: If a smooth affine surface $V$ admits more than one $\C^*$-action up to conjugation and inversion then it belongs to one of the following classes:
(1) it is toric, (2) it is Danilov-Gizatullin, (3) it is special.
If the surface is toric then obviously the subtori of the torus acting on $V$ give many different $\C^*$-action. A Danilov-Gizatullin admits always a finite number of conjugacy classes of $\C^*$-actions. In contrast, a special surface $V$ admits a
continuous family of pairwise non-conjugated $\C^*$-actions depending on one or two parameters. We give an overview of the construction of these surfaces and sketch the proof of this result.