1. 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 $x+y=$
1.【Analysis and Solution】
The original system of equations is transformed into $\left\{\begin{array}{l}(x-1)^{3}+1997(x-1)=-1, \\ (1-y)^{3}+1997(1-y)=-1\end{array}\right.$
Since $f(t)=t^{3}+1997 t$ is monotonically increasing on $(-\infty,+\infty)$ and $f(x-1)=f(1-y)$, it follows that $x-1=1-y$, i.e., $x+y=2$.