Solve X^2+(y-sqrt[3]{x^2})^2=1

$$X ^ { 2 } + ( y - \sqrt[ 3 ] { x ^ { 2 } } ) ^ { 2 } = 1$$

Solve for X (complex solution)

$$X=-\sqrt{-\left(y-x^{\frac{2}{3}}\right)^{2}+1}$$

$$X=\sqrt{-\left(y-x^{\frac{2}{3}}\right)^{2}+1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=0\text{, }&y=-\sqrt{1-X^{2}}\text{ or }y=\sqrt{1-X^{2}}\\x=\left(-\sqrt{1-X^{2}}+y\right)^{3}\text{, }&arg(-\sqrt{1-X^{2}}+y)<\frac{2\pi }{3}\\x=\left(\sqrt{1-X^{2}}+y\right)^{3}\text{, }&arg(\sqrt{1-X^{2}}+y)<\frac{2\pi }{3}\end{matrix}\right.$$

Solve for X

$$X=\sqrt{1-y^{2}+2x^{\frac{2}{3}}y-x^{\frac{4}{3}}}$$

$$X=-\sqrt{1-y^{2}+2x^{\frac{2}{3}}y-x^{\frac{4}{3}}}\text{, }y\geq x^{\frac{2}{3}}-1\text{ and }y\leq x^{\frac{2}{3}}+1$$

Solve for x

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