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