Solve {l}{(1-2a)(3a+1)=x^{3}+2xy-y^{3}}{(x^{2}+xy-y^{2})(x+y)=x^3+2xy-y^2}{(2x-y)(y+4x)+2x(y-3x)=2x^2-y^2}

$$\left. \begin{array} { l } { ( 1 - 2 a ) ( 3 a + 1 ) = x ^ { 3 } + 2 x y - y ^ { 3 } } \\ { ( x ^ { 2 } + x y - y ^ { 2 } ) ( x + y ) = x ^ { 3 } + 2 x y - y ^ { 2 } } \\ { ( 2 x - y ) ( y + 4 x ) + 2 x ( y - 3 x ) = 2 x ^ { 2 } - y ^ { 2 } } \end{array} \right.$$

Solve for a, x, y

$$\left\{\begin{matrix}\\x\leq \frac{15^{\frac{2}{3}}}{6}\text{, }y=0\text{, }a=\frac{\sqrt{25-24x^{3}}+1}{12}\text{; }x\leq \frac{15^{\frac{2}{3}}}{6}\text{, }y=0\text{, }a=\frac{-\sqrt{25-24x^{3}}+1}{12}\text{, }&\text{unconditionally}\\x=\frac{-\sqrt{2y^{2}-2y+1}+1}{2}\text{, }y\in \mathrm{R}\text{, }a=\frac{\sqrt{-3\left(-\sqrt{2y^{2}-2y+1}+1\right)^{3}+24y^{3}+24y\sqrt{2y^{2}-2y+1}-24y+25}+1}{12}\text{; }x=\frac{-\sqrt{2y^{2}-2y+1}+1}{2}\text{, }y\in \mathrm{R}\text{, }a=\frac{-\sqrt{-3\left(-\sqrt{2y^{2}-2y+1}+1\right)^{3}+24y^{3}+24y\sqrt{2y^{2}-2y+1}-24y+25}+1}{12}\text{, }&-24\times \left(\frac{-\sqrt{2y^{2}-2y+1}+1}{2}\right)^{3}+24y^{3}+24y\sqrt{2y^{2}-2y+1}-24y+25\geq 0\\x=\frac{\sqrt{2y^{2}-2y+1}+1}{2}\text{, }y\in \mathrm{R}\text{, }a=\frac{\sqrt{-3\left(\sqrt{2y^{2}-2y+1}+1\right)^{3}+24y^{3}-24y\sqrt{2y^{2}-2y+1}-24y+25}+1}{12}\text{; }x=\frac{\sqrt{2y^{2}-2y+1}+1}{2}\text{, }y\in \mathrm{R}\text{, }a=\frac{-\sqrt{-3\left(\sqrt{2y^{2}-2y+1}+1\right)^{3}+24y^{3}-24y\sqrt{2y^{2}-2y+1}-24y+25}+1}{12}\text{, }&-24\times \left(\frac{\sqrt{2y^{2}-2y+1}+1}{2}\right)^{3}+24y^{3}-24y\sqrt{2y^{2}-2y+1}-24y+25\geq 0\end{matrix}\right.$$