Example 1 Suppose the equation $m x^{2}-(m-2) x+m-3=0$ has integer solutions, try to determine the value of the integer $m$, and find all integer solutions of the equation at this time.
Analysis: If $m=0$, then $2 x-3=0$, in this case the equation has no integer solutions;

When $m \neq 0$, consider $\Delta=-3 m^{2}+8 m+4$. Note that the coefficient of the quadratic term is negative, for the equation to have solutions, then
$-3 m^{2}+8 m+4 \geqslant 0$.
Solving this, we get $\frac{4-2 \sqrt{7}}{3} \leqslant m \leqslant \frac{4+2 \sqrt{7}}{3}$.
Since $m$ is an integer, it can only take the values $1, 2, 3$.
When $m=1$, the equation has solutions: -2 and 1;
When $m=2$, the equation has no integer solutions;
When $m=3$, the equation has integer solutions: 0.
Note: When the coefficient of the quadratic term in the discriminant is negative, solving the inequality yields a finite interval for the parameter. Since the parameter is an integer, the solutions can be discussed accordingly.