$$2 r ( s ^ { 2 } + 1 ) d r + ( r ^ { 4 } + 1 ) d s = 0$$
$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(r=0\text{ and }s=0\right)\text{ or }\left(r=is^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}\text{ and }s\neq 0\right)\text{ or }\left(r=-is^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}\text{ and }s\neq 0\right)\text{ or }\left(r=s^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}-s^{2}-1}\text{ and }s\neq 0\right)\text{ or }\left(r=-s^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}-s^{2}-1}\text{ and }s\neq 0\right)\end{matrix}\right.$
$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(r=0\text{ and }s=0\right)\text{ or }\left(s<0\text{ and }|r|=\sqrt{\frac{\sqrt{s^{4}+s^{2}+1}-1}{s}-s}\right)\text{ or }\left(s<0\text{ and }|r|=\sqrt{-\frac{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}{s}}\right)\end{matrix}\right.$
$\left\{\begin{matrix}r=s^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}-s^{2}-1}\text{; }r=-s^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}-s^{2}-1}\text{; }r=-is^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}\text{; }r=is^{-\frac{1}{2}}\sqrt{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}\text{, }&s\neq 0\\r=0\text{, }&s=0\\r\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$
$\left\{\begin{matrix}r=-\sqrt{-\frac{-\sqrt{s^{4}+s^{2}+1}+s^{2}+1}{s}}\text{; }r=\sqrt{-\frac{-\sqrt{s^{4}+s^{2}+1}+s^{2}+1}{s}}\text{; }r=\sqrt{-\frac{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}{s}}\text{; }r=-\sqrt{-\frac{\sqrt{s^{4}+s^{2}+1}+s^{2}+1}{s}}\text{, }&s<0\\r=0\text{, }&s=0\\r\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$