Solve z(2z+2y-z)+y(2y-2)-1=x(x-2)+y(y-2)+z(z-2)

$$z ( 2 z + 2 y - z ) + y ( 2 y - 2 ) - 1 = x ( x - 2 ) + y ( y - 2 ) + z ( z - 2 )$$

Solve for x (complex solution)

$$x=\sqrt{y^{2}+2yz+2z}+1$$

$$x=-\sqrt{y^{2}+2yz+2z}+1$$

Solve for y (complex solution)

$$y=\sqrt{x^{2}-2x+z^{2}-2z+1}-z$$

$$y=-\sqrt{x^{2}-2x+z^{2}-2z+1}-z$$

Solve for x

$$x=\sqrt{y^{2}+2yz+2z}+1$$

$$x=-\sqrt{y^{2}+2yz+2z}+1\text{, }\left(y\leq -1\text{ or }z\geq -\frac{y^{2}}{2y+2}\right)\text{ and }\left(y\geq -1\text{ or }z\leq -\frac{y^{2}}{2y+2}\right)$$

Solve for y

$$y=\sqrt{x^{2}-2x+z^{2}-2z+1}-z$$

$$y=-\sqrt{x^{2}-2x+z^{2}-2z+1}-z\text{, }z\geq \sqrt{2x-x^{2}}+1\text{ or }z\leq -\sqrt{2x-x^{2}}+1\text{ or }x\leq 0\text{ or }x\geq 2$$