Finding distributions on $\mathbb{R^2}$ constrained by 2 equations. The question asks us to classify all distributions in $\mathbb{R}^2$, say $u$, such that $$(xy)u(x,y) = (x^2-y^2)u(x,y)= 0.$$
I am genuinely stuck, any help is extremely appreciated.
 A: Solving $xy \, u(x,y)=0$:
We know that $x\,v(x)=0$ has solutions $v(x)=C\,\delta(x),$ where $C$ is a constant. Generalizing this we get that $y\,u(x,y) = A(y)\,\delta(x)$ for some distribution $A(y).$ Then $u(x,y) = B(y)\,\delta(x) + C(x)\,\delta(y),$ where $A(y)=y\,B(y)$ and $C(x)$ is some distribution. Thus,
$$
u(x,y) = \delta(x) \otimes B(y) + C(x) \otimes \delta(y),
$$
where $B(y)$ and $C(x)$ are some distributions.
We want to find solutions that also satisfy $(x^2-y^2)\,u(x,y)=0,$ i.e.
$$
0 
= (x^2-y^2)(\delta(x) \otimes B(y) + C(x) \otimes \delta(y))
= \delta(x) \otimes (-y^2)\,B(y) + x^2\,C(x) \otimes \delta(y)
$$
since $x^2\,\delta(x)=0=y^2\,\delta(y).$
Thus we shall have $\delta(x) \otimes y^2\,B(y) = x^2\,C(x) \otimes \delta(y)$ meaning that
$$
\begin{cases}
\delta(x) = \lambda \, x^2\,C(x) \\
y^2\,B(y) = \lambda^{-1} \delta(y) \\
\end{cases}
$$
for some constant $\lambda.$
The solutions to these two equations are,
$$\begin{align}
B(y) &= \frac12 \lambda^{-1}\, \delta''(y) + E\,\delta'(y) + F\,\delta(y) \\
C(x) &= \frac12 \lambda \,\delta''(x) + G\,\delta'(x) + H\,\delta(x) \\
\end{align}$$
where $E,F,G,H$ are constants.
Thus,
$$
u(x,y) 
= \delta(x) \otimes (\frac12 \lambda^{-1}\, \delta''(y) + E\,\delta'(y) + F\,\delta(y)) 
+ (\frac12 \lambda \,\delta''(x) + G\,\delta'(x) + H\,\delta(x)) \otimes \delta(y)
.
$$
