  Let $\rho$ be an algebraic action of the additive group $\mathbb{C}^+$ on the
three-dimensional affine space $\mathbb{C}^3$. We describe the group
$\textrm{Cent}(\rho)$ of polynomial automorphisms of $\mathbb{C}^3$ that
commute with $\rho$. A particular emphasis lies in the description of the
automorphisms in $\textrm{Cent}(\rho)$ coming from algebraic
$\mathbb{C}^+$-actions. As an application we prove that the automorphisms in
$\textrm{Cent}(\rho)$ that are the identity on the algebraic quotient of $\rho$
form a characteristic subgroup of $\textrm{Cent}(\rho)$.
