$\left\{\begin{matrix}a=\frac{2y^{2}+\sqrt{8y^{2}+1}+1}{2xy^{2}}\text{, }&x\neq 0\text{ and }y\neq 0\text{ and }arg(y-axy)<\pi \text{ and }arg(\frac{\sqrt{8y^{2}+1}+1}{y})\geq \pi \\a=-\frac{3}{x}\text{, }&arg(\frac{-\sqrt{2}iax+\sqrt{2}i}{4})<\pi \text{ and }y=\frac{\sqrt{2}i}{4}\text{ and }x\neq 0\\a=-\frac{1}{x}\text{, }&x\neq 0\text{ and }y=0\\a\in \mathrm{C}\text{, }&y=1\text{ and }x=0\\a=-\frac{-2y^{2}+\sqrt{8y^{2}+1}-1}{2xy^{2}}\text{, }&x\neq 0\text{ and }y\neq 0\text{ and }arg(y-axy)<\pi \text{ and }arg(\frac{\sqrt{8y^{2}+1}-1}{y})<\pi \end{matrix}\right.$

$\left\{\begin{matrix}x=\frac{2y^{2}+\sqrt{8y^{2}+1}+1}{2ay^{2}}\text{, }&a\neq 0\text{ and }y\neq 0\text{ and }arg(y-axy)<\pi \text{ and }arg(\frac{\sqrt{8y^{2}+1}+1}{y})\geq \pi \\x=-\frac{3}{a}\text{, }&arg(\frac{-\sqrt{2}iax+\sqrt{2}i}{4})<\pi \text{ and }y=\frac{\sqrt{2}i}{4}\text{ and }a\neq 0\\x=-\frac{1}{a}\text{, }&a\neq 0\text{ and }y=0\\x\in \mathrm{C}\text{, }&y=1\text{ and }a=0\\x=-\frac{-2y^{2}+\sqrt{8y^{2}+1}-1}{2ay^{2}}\text{, }&a\neq 0\text{ and }y\neq 0\text{ and }arg(y-axy)<\pi \text{ and }arg(\frac{\sqrt{8y^{2}+1}-1}{y})<\pi \end{matrix}\right.$

$\left\{\begin{matrix}a=\frac{2y^{2}+\sqrt{8y^{2}+1}+1}{2xy^{2}}\text{, }&y<0\text{ and }x\neq 0\\a=-\frac{-2y^{2}+\sqrt{8y^{2}+1}-1}{2xy^{2}}\text{, }&y>0\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&y=1\text{ and }x=0\\a=-\frac{1}{x}\text{, }&x\neq 0\text{ and }y=0\end{matrix}\right.$

$\left\{\begin{matrix}x=\frac{2y^{2}+\sqrt{8y^{2}+1}+1}{2ay^{2}}\text{, }&y<0\text{ and }a\neq 0\\x=-\frac{-2y^{2}+\sqrt{8y^{2}+1}-1}{2ay^{2}}\text{, }&y>0\text{ and }a\neq 0\\x\in \mathrm{R}\text{, }&y=1\text{ and }a=0\\x=-\frac{1}{a}\text{, }&a\neq 0\text{ and }y=0\end{matrix}\right.$