  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.
