  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 $xy = \varphi(z)$, $x^ny=\varphi(Z)$, and
$f(x)x = \varphi(z)$.
