Autour d'une surface rationnelle dans ${\mathbb{C}}^3$

Affine surfaces in ${\mathbb{C}}^3$ defined by an equation of the form $x^{n}z-Q(x,y)=0$ 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 ${\mathbb{C}}_{+}$ 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 ${\mathbb{C}}^3$ 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 ${\mathbb{C}}_{+}$-action induced by an action on ${\mathbb{C}}^3$. We give an elementary and self-contained proof of this fact. As an application, we construct a wild but stably-tame automorphisme of ${\mathbb{C}}^3$ which seems to be new.