Autour d'une surface rationnelle dans C³

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Affine surfaces in C³ defined by an equation of the form xⁿz-Q(x,y)=0 have been increasingly studied during the past 15 years. Of particular interest is the fact that they come equipped with an action of the additive group C₊ induced by such an action on the ambient space. The litterature of the last decade may lead one to believe that there are essentially no other of rational surfaces in C³ with this property. In this note, we construct an explicit example of a surface nonisomorphic to a one of the above type but equipped with a free C₊-action induced by an action on C³. We give an elementary and self-contained proof of this fact. As an application, we construct a wild but stably-tame automorphisme of C³ which seems to be new.