On the isotropy group of a simple derivation

Abstract: Let $R=K[X_1,\dots, X_n]$ be a polynomial ring in $n$ variables over a field $K$ of charactersitic zero and $d$ a $K$-derivation of $R$. Consider the isotropy group if $d$: $ \text{Aut}(R)_d :={\rho \in \text{Aut}_K(R)|\; \rho d \rho^{-1}=d}$. In his doctoral thesis, Baltazar proved that if $d$ is a simple Shamsuddin derivation of $K[X_1,X_2]$, then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan generalized this result to an arbitrary derivation of $K[X_1,X_2]$ proving that a derivation of $K[X_1,X_2]$ is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring $R=K[X_1,\dots, X_n]$ is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of $K[X_1,X_2]$ and prove that they are finite cyclic groups.