Solve x^2+xy+y^2=x^2y^2

$${ x }^{ 2 } +xy+ { y }^{ 2 } = { x }^{ 2 } { y }^{ 2 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{y^{2}\left(4y^{2}-3\right)}-y}{2\left(1-y^{2}\right)}\text{; }x=-\frac{\sqrt{y^{2}\left(4y^{2}-3\right)}+y}{2\left(1-y^{2}\right)}\text{, }&y\neq 1\text{ and }y\neq -1\\x=-y\text{, }&y=1\text{ or }y=-1\end{matrix}\right.$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{\sqrt{x^{2}\left(4x^{2}-3\right)}-x}{2\left(1-x^{2}\right)}\text{; }y=-\frac{\sqrt{x^{2}\left(4x^{2}-3\right)}+x}{2\left(1-x^{2}\right)}\text{, }&x\neq 1\text{ and }x\neq -1\\y=-x\text{, }&x=1\text{ or }x=-1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{y\left(\sqrt{4y^{2}-3}-1\right)}{2\left(1-y^{2}\right)}\text{; }x=-\frac{y\left(\sqrt{4y^{2}-3}+1\right)}{2\left(1-y^{2}\right)}\text{, }&\left(y\neq 1\text{ and }y\geq \frac{\sqrt{3}}{2}\right)\text{ or }\left(y\neq -1\text{ and }y\leq -\frac{\sqrt{3}}{2}\right)\\x=-y\text{, }&y=0\text{ or }|y|=1\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=\frac{x\left(\sqrt{4x^{2}-3}-1\right)}{2\left(1-x^{2}\right)}\text{; }y=-\frac{x\left(\sqrt{4x^{2}-3}+1\right)}{2\left(1-x^{2}\right)}\text{, }&\left(x\neq 1\text{ and }x\geq \frac{\sqrt{3}}{2}\right)\text{ or }\left(x\neq -1\text{ and }x\leq -\frac{\sqrt{3}}{2}\right)\\y=-x\text{, }&x=0\text{ or }|x|=1\end{matrix}\right.$$