Example 2. It is known that the equation $a x^{2}-(a-3) x+a-2=0$ has at least one integer solution, find the integer $a$.

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Example 2. It is known that the equation $a x^{2}-(a-3) x+a-2=0$ has at least one integer solution, find the integer $a$.
Touch If $a=0$, the original equation becomes $3 x-2=0$, which has no integer solutions, so $a \neq 0$.
$$
\begin{array}{l}
\because \Delta=(a-3)^{2}-4 a(a-2) \geqslant 0, \\
\therefore 3 a^{2}-2 a-9 \leqslant 0, \\
\therefore \frac{1-2 \sqrt{7}}{3} \leqslant a \leqslant \frac{1+2 \sqrt{7}}{3} .
\end{array}
$$

The integer values of $a$ that satisfy the above inequality are $-1, 1, 2$.
Substituting these values of $a$ into the original equation for verification, when $a=-1$, $x=1,3$ (two integer solutions). When $a=2$, $x=0$, $\frac{1}{2}$ (one integer solution). Therefore, the integer values of $a$ that meet the requirements are -1, 2.