Example 2. Find the integer solutions of the equation
$$
2 x y+2 x z-x y z=4 z
$$

for the unknowns $x, y, z$ satisfying the conditions $x \geq 1, y=3, z=3$.
Factorizing the left side of equation (1), we get
$$
x(2 y+2 z-y z)=4 z .
$$

Since $x \geqslant 1, z \geqslant 3$, it follows that $2 y+2 z-y z>0$, i.e., $\square$
$$
y z-2 y-2 z<0 \text {. }
$$
(3) can be transformed into $(y-2)(z-2)<4$.

Since $y \geqslant 3, z \geqslant 3$, we know that $y-2 \geqslant 1, z-2 \geqslant 1$. Therefore, $(y-2)(z-2) \geq 1$.
From (4) and (5), we get
$$
1 \leq(3-2)(z-2)<4 .
$$

We will discuss three cases: $(y-2)(z-2)=1$, $(y-2)(z-2)=2$, and $(y-2)(z-2)=3$, to obtain all the integer solutions of equation (1)
$$
(x, y, z)=(4,3,3),(6,4,3),(8,3 \text {, }
$$
4), $(12,5,3) .(20,3,5)$.