Example 2. Find a pair of integers satisfying
$$
\left\{\begin{array}{l}
y-\left|x^{2}-2 x\right|+\frac{1}{2}>0 \\
y+|x-1|<2
\end{array}\right.
$$
Solving the original system of inequalities can be transformed into
$$
\left\{\begin{array}{l}
-y-\frac{1}{2}<x^{2}-2 x<y+\frac{1}{2} \\
-2+y<x-1<2-y \\
-y+\frac{1}{2}<(x-1)^{2}<y+\frac{3}{2} \\
-1+y<x<3-y
\end{array}\right.
$$

That is,

Draw the solution region of the system of inequalities as shown by the shaded area in Figure 2. The integer points in this region are $(0,0)$, $(2,0)$, and $(1,1)$. Therefore, the integer solutions to the system of inequalities are $\left\{\begin{array}{l}x=0, \\ y=0,\end{array} \quad\left\{\begin{array}{l}x=2, \\ y=1 ;\end{array}\right.\right.$ $\left\{\begin{array}{l}x=1, \\ y=1\end{array}\right.$.