What can be said about pairs of polynomials $P, Q\in\mathbb{C}[x, y, z]$, such that $\frac{\partial P}{\partial y}\frac{\partial Q}{\partial z} - \frac{\partial P}{\partial z}\frac{\partial Q}{\partial y}\in\mathbb{C}[x]$?

I am interested in examples of such $P, Q$.

The similar question can be asked if we replace $\mathbb{C}[x]$ with any field $F$ of characteristic zero and $\mathbb{C}[x, y, z]$ with $F[y, z]$.