$4 \cdot 145$ Solve the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+x_{3}=6, \\
x_{2}+x_{3}+x_{4}=9, \\
x_{3}+x_{4}+x_{5}=3, \\
x_{4}+x_{5}+x_{6}=-3, \\
x_{5}+x_{6}+x_{7}=-9, \\
x_{6}+x_{7}+x_{8}=-6, \\
x_{7}+x_{8}+x_{1}=-2, \\
x_{8}+x_{1}+x_{2}=2 .
\end{array}\right.
$$
[Solution] Add all 8 equations, we get
$$
\begin{aligned}
& 3\left(x_{1}+x_{2}+x_{3}+\cdots+x_{8}\right)=0, \\
\therefore \quad & x_{1}+x_{2}+x_{3}+\cdots+x_{8}=0 .
\end{aligned}
$$
(1) + (4) + (7) - (9), we get $x_{1}=1$.

Similarly, we can get $x_{2}=2, x_{3}=3, x_{4}=4, x_{5}=-4$,
$$
x_{6}=-3, x_{7}=-2, x_{8}=-1 .
$$