Automorphisms of ${\mathbb{C}}^3$ Commuting with a ${\mathbb{C}}^{+}$-Action

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 $\text{Cent}(\rho )$ of polynomial automorphisms of ${\mathbb{C}}^3$ that commute with $\rho$. A particular emphasis lies in the description of the automorphisms in $\text{Cent}(\rho )$ coming from algebraic ${\mathbb{C}}^{+}$-actions. As an application we prove that the automorphisms in $\text{Cent}(\rho )$ that are the identity on the algebraic quotient of $\rho$ form a characteristic subgroup of $\text{Cent}(\rho )$.