Let $\displaystyle A=\mathbb{C}[x,y,z]$ with the relations $\displaystyle yx=xy+1, \ yz=zy, \ xz=zx.$

: for every $\displaystyle a \in A$ there exists $\displaystyle b \in A \setminus \mathbb{C}$ such that $\displaystyle \displaystyle \frac{\partial b}{\partial z}=ab-ba.$

True or False

the problem is made up by me and i have some reasons (no proof) to believe that it is true!

note that if at least one of three variables $\displaystyle x,y,z$ does not appear in $\displaystyle a,$ then the statement is trivially true.

this could be a good undergraduate project - or maybe it's too hard?