Solve {left(frac{1/x^2-3y^2right)}^3}{4pir^3}-z=0

$$\frac{ { \left( \frac{ 1 }{ { x }^{ 2 } } -3 { y }^{ 2 } \right) }^{ 3 } }{ 4 \pi { r }^{ 3 } } -z=0$$

Solve for r

$$\left\{\begin{matrix}r=\frac{4^{\frac{2}{3}}\left(1-3\left(xy\right)^{2}\right)}{4\sqrt[3]{\pi z}x^{2}}\text{, }&\left(|x|\neq \frac{\sqrt{3}}{3|y|}\text{ or }y=0\right)\text{ and }x\neq 0\text{ and }z\neq 0\\r\neq 0\text{, }&z=0\text{ and }y\neq 0\text{ and }|x|=\frac{\sqrt{3}}{3|y|}\text{ and }x\neq 0\end{matrix}\right.$$

Solve for x

$$x=\frac{1}{\sqrt{3y^{2}+\sqrt[3]{4\pi z}r}}$$

$$x=-\frac{1}{\sqrt{3y^{2}+\sqrt[3]{4\pi z}r}}\text{, }\left(z>0\text{ and }r>0\right)\text{ or }\left(z<0\text{ and }r<0\right)\text{ or }\left(|y|>\frac{\sqrt{3}\sqrt[3]{2}\sqrt[6]{-\pi zr^{3}}}{3}\text{ and }z\leq 0\text{ and }r>0\right)\text{ or }\left(|y|>\frac{\sqrt{3}\sqrt[3]{2}\sqrt[6]{-\pi zr^{3}}}{3}\text{ and }z\geq 0\text{ and }r<0\right)$$