On the Isotropy Groups of Non-Invertible Simple Derivations

Let $k$ be a field of characteristic zero, and let $i$ and $n$ be positive integers with $i\ge 2$ and $n>i$. Consider a non-invertible $k$-derivation $d_i$ of the polynomial ring $k[x_1,\dots ,x_i]$. Let $d_n$ be an extension of $d_i$ to a derivation of $k[x_1,\dots ,x_n]$ such that $d_n(x_j)\in k[x_{j-1}]\setminus k$ for each $j$ with $i+1\le j\le n$. In this article, we undertake a systematic study of the isotropy groups associated with such non-invertible derivations. We establish sufficient conditions on $d_i$ under which the isotropy group of the non-invertible simple derivation $d_n$ is conjugate to a subgroup of translations.