Simple derivations and isotropy on Danielewski-type algebras

Let $\text{K}$ be an algebraically closed field of characteristic zero. We study isotropy groups of simple derivations on Danielewski-type algebras $$A_{c,q}=\text{K}[x,y,z]\mathrm{/}(c(x)z-q(x,y))\mathrm{.}$$ More precisely, motivated by the recent result of Mendes and Pan for simple derivations of $\text{K}[x,y]$ (\cite[Theorem 1]{MP}), we ask whether every simple derivation $D$ of $A_{c,q}$ has trivial isotropy group. We prove that this property holds generically: for each fixed pair of degrees $\mathrm{°}(c)\ge 2$ and ${\mathrm{°}}_y(q)\ge 2$, there exists a nonempty Zariski open subset of the parameter space of reduced pairs $(c,q)$ such that every simple derivation of $A_{c,q}$ has trivial isotropy group. On the other hand, we construct a special Danielewski-type algebra admitting a simple derivation with nontrivial isotropy. Thus, the Mendes-Pan phenomenon holds generically for Danielewski-type algebras, but fails in full generality. Over $\text{K}=\mathbb{C}$, we also discuss the associated foliations and formulate questions about their polynomial and birational symmetries.