  The polynomial automorphisms of the affine plane have been studied a lot: if
$f$ is such an automorphism, then either $f$ preserves a rational fibration,
has an uncountable centralizer and its first dynamical degree equals $1$, or
$f$ preserves no rational curves, has a countable centralizer and its first
dynamical degree is $>1$. In higher dimensions there is no such description. In
this article we study a family $(\Psi_\alpha)_\alpha$ of polynomial
automorphisms of $\mathbb{C}^3$. We show that the first dynamical degree of
$\Psi_\alpha$ is $>1$, that $\Psi_\alpha$ preserves a unique rational fibration
and has an uncountable centralizer. We then describe the dynamics of the family
$(\Psi_\alpha)_\alpha$, in particular the speed of points escaping to infinity.
We also observe different behaviors according to the value of the parameter
$\alpha$.
