Integer solutions of $xy+xz+yz-2xyz=0$

Question

I have to find the positive integer solutions of the equation $$xy+xz+yz-2xyz=0.$$ Note: If there are solutions, they should be finite in number because $xyz$ is of third degree.

Answer 1

If wlog $x=0$, then $(x,y,z)=(0,0,z),(0,y,0),y,z,\in\Bbb Z_{\ge 0}$ are all the solutions.

Wlog $x\ge y\ge z\ge 1$.

$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2\le \frac{3}{z}\implies z\le 1.5\implies z=1$$

$$\frac{1}{x}+\frac{1}{y}=1\iff (x-1)(y-1)=1\iff x=y=2$$

Answer: $(x,0,0),(0,y,0),(0,0,z), x,y,z\in\Bbb Z_{\ge 0}, (x,y,z)=(2,2,1),(2,1,2),(1,2,2)$ are all the solutions.