II. (20 points) Find all positive integer solutions $(x, y, z)$ of the indefinite
equation
$$
2 x^{2}+6 x y-12 x z+x-3 y+6 z-6=0
$$
Let $u=y-2z$, then we have
$$
2 x^{2}+6 x u+x-3 u-6=0,
$$

which is equivalent to $(2 x-1)(x+3 u+1)=5$.
Thus, we should have
$$
\left\{\begin{array}{l}
2 x-1=5 \\
x+3 u+1=1
\end{array}\right.
$$
(1) or $\left\{\begin{array}{l}2 x-1=1, \\ x+3 u+1=5 .\end{array}\right.$

Solving (1), we get $\left\{\begin{array}{l}x=3, \\ u=-1,\end{array}\right.$ which means $x=3, y=2 t-1, z=t$.
Solving (2), we get $\left\{\begin{array}{l}x=1, \\ u=1,\end{array}\right.$ which means $x=1, y=2 t+1, z=t$.
$t$ is any positive integer.