Solve x^3+y^3=3xy

$${ x }^{ 3 } + { y }^{ 3 } =3xy$$

Solve for x

$$\left\{\begin{matrix}x=0\text{, }&y=0\\x=-\frac{2^{\frac{2}{3}}\left(1+\sqrt{3}i\right)\left(\sqrt{y^{3}\left(y^{3}-4\right)}-y^{3}\right)^{-\frac{1}{3}}\left(\left(-\sqrt{3}i-1\right)\left(\sqrt{y^{3}\left(y^{3}-4\right)}-y^{3}\right)^{\frac{2}{3}}+2\times 2^{\frac{2}{3}}y\right)}{8}\text{; }x=\frac{2^{\frac{2}{3}}\left(\sqrt{y^{3}\left(y^{3}-4\right)}-y^{3}\right)^{-\frac{1}{3}}\left(\left(\sqrt{y^{3}\left(y^{3}-4\right)}-y^{3}\right)^{\frac{2}{3}}+2^{\frac{2}{3}}y\right)}{2}\text{; }x=-\frac{2^{\frac{2}{3}}\left(-\sqrt{3}i+1\right)\left(\sqrt{y^{3}\left(y^{3}-4\right)}-y^{3}\right)^{-\frac{1}{3}}\left(\left(-1+\sqrt{3}i\right)\left(\sqrt{y^{3}\left(y^{3}-4\right)}-y^{3}\right)^{\frac{2}{3}}+2\times 2^{\frac{2}{3}}y\right)}{8}\text{, }&y\neq 0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=0\text{, }&x=0\\y=-\frac{2^{\frac{2}{3}}\left(1+\sqrt{3}i\right)\left(\sqrt{x^{3}\left(x^{3}-4\right)}-x^{3}\right)^{-\frac{1}{3}}\left(\left(-\sqrt{3}i-1\right)\left(\sqrt{x^{3}\left(x^{3}-4\right)}-x^{3}\right)^{\frac{2}{3}}+2\times 2^{\frac{2}{3}}x\right)}{8}\text{; }y=\frac{2^{\frac{2}{3}}\left(\sqrt{x^{3}\left(x^{3}-4\right)}-x^{3}\right)^{-\frac{1}{3}}\left(\left(\sqrt{x^{3}\left(x^{3}-4\right)}-x^{3}\right)^{\frac{2}{3}}+2^{\frac{2}{3}}x\right)}{2}\text{; }y=-\frac{2^{\frac{2}{3}}\left(-\sqrt{3}i+1\right)\left(\sqrt{x^{3}\left(x^{3}-4\right)}-x^{3}\right)^{-\frac{1}{3}}\left(\left(-1+\sqrt{3}i\right)\left(\sqrt{x^{3}\left(x^{3}-4\right)}-x^{3}\right)^{\frac{2}{3}}+2\times 2^{\frac{2}{3}}x\right)}{8}\text{, }&x\neq 0\end{matrix}\right.$$