This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some third element $H$?"
It was answered in the negative (the counterexample seems to be due to Dixmier and Makar-Limanov).
Now replace the field of characteristic zero $k$ by a non-normal integral domain of characteristic zero $R$, for example $R=\mathbb{C}[t^2,t^3]$.
Is it possible to adjust $P$ and $Q$ to some $\tilde{P}$ and $\tilde{Q}$ satisfying the following three conditions:
(i) $[\tilde{P},\tilde{Q}]=0$.
(ii) $\tilde{P}$ and $\tilde{Q}$ are not polynomials in some third element $H$.
(iii) $\tilde{P}$ has a 'Dixmier mate', namely, there exists $B$ in the first Weyl algebra over $R$ such that $[\tilde{P},B]=1$.
Remarks:
(1) The analogous question in $R[x,y]$ can be found here.
(2) In contrast, in $k[x,y]$, it is true that if $\operatorname{Jac}(p,q)=0$, then $p=u(h)$ and $q=v(h)$, for some $h \in k[x,y]$, $u(T),v(T) \in k[T]$; see, for example, Corollary 1.3 in Nagata's paper ($k$ is an algebraically closed field of characteristic zero) or this question ($k$ is an arbitrary field). $k$ can be replaced by any normal integral domain of characteristic zero, see this answer.
Any hints and comments are welcome!