Example 2 Let $x, y$ be real numbers, and satisfy
$$
\left\{\begin{array}{l}
(x-1)^{3}+1997(x-1)=-1, \\
(y-1)^{3}+1997(y-1)=1 .
\end{array}\right.
$$

Then $x+y=$ $\qquad$
(1997, National High School Mathematics League Competition)
【Analysis】By comparing the forms on the left side of the two equations, we can construct the function $f(x)=x^{3}+1997 x$.

This is both an odd function and a monotonically increasing function. The system of equations becomes
$$
\begin{array}{l}
\left\{\begin{array}{l}
f(x-1)=-1, \\
f(y-1)=1
\end{array}\right. \\
\Rightarrow f(x-1)=-f(y-1)=f(1-y) \\
\Rightarrow x-1=1-y \Rightarrow x+y=2 .
\end{array}
$$