On Isotropy Group of Danielewski Surfaces

Abstract

In the present work we consider differential rings of the form $(\mathcal{B},D)$ where $\mathcal{B}$ is a Danielewski surface and $D$ is a locally nilpotent derivation on $\mathcal{B}$. Influenced by several recent works, we describe the isotropy group of a locally nilpotent derivation, $D$, on Danielewski surfaces, in the cases $xy=\phi (z)$, $x^{n}y=\phi (Z)$, and $f(x)x=\phi (z)$.