Define $a\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\clubsuit y=y\clubsuit x$?
$\textbf{(A)}\ \text{a finite set of points}\\ \qquad\textbf{(B)}\ \text{one line}\\ \qquad\textbf{(C)}\ \text{two parallel lines}\\ \qquad\textbf{(D)}\ \text{two intersecting lines}\\ \qquad\textbf{(E)}\ \text{three lines}$
$x\clubsuit y = x^2y-xy^2$ and $y\clubsuit x = y^2x-yx^2$. Therefore, we have the equation $x^2y-xy^2 = y^2x-yx^2$ Factoring out a $-1$ gives $x^2y-xy^2 = -(x^2y-xy^2)$ Factoring both sides further, $xy(x-y) = -xy(x-y)$. It follows that if $x=0$, $y=0$, or $(x-y)=0$, both sides of the equation equal 0. By this, there are 3 lines ($x=0$, $y=0$, or $x=y$) so the answer is $\boxed{\textbf{(E)}\text{ three lines}}$.