Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: $\tilde{D}_{X}(f)=f_{x}*P+f_{y}*Q$ or $\tilde{D}_{X}(f)=P*f_{x}+Q*f_{y}$ where $*$ is the Moyal product. The lat...