$\left\{\begin{matrix}y=-\frac{z-1000}{x^{2}+xz-x-z-1}\text{, }&x=1\text{ or }z\neq -\frac{x^{2}-x-1}{x-1}\\y\in \mathrm{C}\text{, }&\left(x=\frac{-\sqrt{1002005}-999}{2}\text{ or }x=\frac{\sqrt{1002005}-999}{2}\right)\text{ and }z=1000\end{matrix}\right.$

$\left\{\begin{matrix}y=-\frac{z-1000}{x^{2}+xz-x-z-1}\text{, }&x=1\text{ or }z\neq -\frac{x^{2}-x-1}{x-1}\\y\in \mathrm{R}\text{, }&\left(x=\frac{-\sqrt{1002005}-999}{2}\text{ or }x=\frac{\sqrt{1002005}-999}{2}\right)\text{ and }z=1000\end{matrix}\right.$

$\left\{\begin{matrix}x=\frac{\sqrt{y\left(yz^{2}+2yz+5y-4z+4000\right)}-yz+y}{2y}\text{; }x=\frac{-\sqrt{y\left(yz^{2}+2yz+5y-4z+4000\right)}-yz+y}{2y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&z=1000\text{ and }y=0\end{matrix}\right.$

$\left\{\begin{matrix}x=\frac{\sqrt{y\left(yz^{2}+2yz+5y-4z+4000\right)}-yz+y}{2y}\text{; }x=\frac{-\sqrt{y\left(yz^{2}+2yz+5y-4z+4000\right)}-yz+y}{2y}\text{, }&\left(y\geq \frac{\sqrt{1002005}-1001}{2}\text{ or }y<0\text{ or }z\leq -\frac{4\sqrt{1-1001y-y^{2}}+2y-4}{2y}\text{ or }z\geq \frac{4\sqrt{1-1001y-y^{2}}-2y+4}{2y}\right)\text{ and }\left(y\leq \frac{-\sqrt{1002005}-1001}{2}\text{ or }y>0\text{ or }z\geq -\frac{4\sqrt{1-1001y-y^{2}}+2y-4}{2y}\text{ or }z\leq \frac{4\sqrt{1-1001y-y^{2}}-2y+4}{2y}\right)\text{ and }y\neq 0\\x\in \mathrm{R}\text{, }&z=1000\text{ and }y=0\end{matrix}\right.$