Example 9. Find all integer values of $x$ that make $y=\frac{x^{2}-2 x+4}{x^{2}-3 x+3}$ an integer.
Solve $y=1+\frac{x+1}{x^{2}-3 x+3}$.
$$
\text { Let } \frac{x+1}{x^{2}-3 x+3}=k \text {, }
$$

i.e., $k x^{2}-(3 k+1) x+3 k-1=0, x$ is a real number.
$\Delta(k) \geqslant 0$,

i.e., $(3 k+1)^{2}-4 k(3 k-1) \geqslant 0$.
Thus, $3 k^{2}-10 k-1 \leqslant 0$.
$\frac{5-\sqrt{28}}{3} \leqslant k \leqslant \frac{5+\sqrt{28}}{3}$.
$\because y \in Z$, so $k \in Z$,
$\therefore k=0,1,2,3$.
Substituting and verifying, when $k=0$, $x_{1}=-1$; when $k=2$,
$$
x_{2}=1, \quad k=3 \text { when }, \quad x_{3}=2 \text {. }
$$