Example 1. For what integer values of $m$ does the equation $2 x^{2}+(1-2 m) x+m^{2}-2 m=0$ have integer solutions?
Solve $\because \Delta=-4 m^{2}+12 m+1$,
$$
\begin{array}{l}
\therefore x=-\frac{2 m-1 \pm \sqrt{\Delta}}{4} . \\
\because x, m \text { are integers, } \\
\therefore \Delta=-4 m^{2}+12 m+1 \geqslant 0 .
\end{array}
$$

Solving, we get $\frac{3-\sqrt{10}}{2} \leqslant m \leqslant \frac{3+\sqrt{10}}{2}$.
Thus, $m=0,1,2,3$.
Substituting the values of $m$ into (1), the integer solutions of the equation are $0,1,0,1$ respectively.