Extension of automorphisms of rational smooth affine curves

We provide the existence, for every complex rational smooth affine curve Γ, of a linear action of Aut(Γ) on the affine 3-dimensional space A³, together with a Aut(Γ)-equivariant closed embedding of Γ into A³. It is not possible to decrease the dimension of the target, the reason for this obstruction is also precisely described.