Given the system of ODEs, $$x'=x^3-3xy^2$$ $$y'=3x^2 y-y^3,$$ it can be shown that the system may be written as $z'=z^3$, where $z=x+iy$. However, I don't seem to get how to show that $\Im m\{\frac{1}{z^2}\}$ is constant. Moreover, this fact can somehow be used to sketch the phase portrait.
The "best" I've so far come up with is this:
$(\frac{1}{z^2})'=-\frac{2z'}{z^3}=-2 \implies \frac{1}{z^2}=-2t$
We can also use polar coordinates, but the question is where and how exactly?
I'd appreciate some clarification. This is truly new to me.