Example 2. For what integer values of $m$, does the equation $m x^{2}+(m-2) x+m-3=0$ ($m \neq 0$) have integer solutions? And find these solutions.
Solving by imitating Example 1, we get: when $m=1$, the equation has integer solutions $x=2, -1$;  
when $m=3$, the equation has an integer solution $x=0$.  
Summary $1^{\circ}$ When $\Delta=A m^{2}+B m+C$ with $A<0$, let $\Delta \geqslant 0$, solving yields $\alpha \leqslant m \leqslant \beta$, determine the integers $m$ within the interval $[\alpha, \beta]$, then find the solutions of the equation corresponding to each $m$, thereby determining the $x$ that meets the problem's requirements.