Problem: Find all distributions $f$ in $\mathcal D'(\mathbb R^2)$ so that $xyf=0$ and $(x^2-y^2)f=0.$
My Thoughts: Note that $c\delta_{(0,0)}$ where $c\in\mathbb C$ satisfies the distributional equations above. It appears that $\operatorname{supp}(u)=\{x=0\}\cap\{y=0\}=(0,0)$ which would imply that $u$ is a finite linear combination of $\delta_0$ and its derivatives.
Is my reasoning on the right track? Apologies if I am not showing enough details, I am trying to make sure that my intuition is on point.
Thank you for your time.