Example 11. Find the positive integer solutions of the system of equations
$$
\left\{\begin{array}{l}
6 x-y-z=18, \\
x^{2}+y^{2}+z^{2}=1987
\end{array}\right.
$$
$$
\begin{array}{l}
\text { Sol } \because x, y, \quad z \in N, \\
\therefore(y+z)^{2} \leqslant 2\left(y^{2}+z^{2}\right)=2\left(1987-x^{2}\right) \\
y^{2}+z^{2}=1987-x^{2}>1987-14^{2} \\
=1791 \geqslant 42^{2}, \\
\quad y+z \geqslant 42 .
\end{array}
$$
- From (1), $6 x=18+(y+z) \geqslant 60$, we have $x \geqslant 10$. Therefore, $10 \leqslant x<14$.

And from the analysis of (1) and (2), $y+z$ must be an even number, so $y^{2}+z^{2}$ must also be an even number.

From (2), we know that $x$ must be an odd number, thus, $x=11,13$. When $x=11$, there are no integer solutions, so it should be discarded; when $x=13$, substituting into the original system of equations yields
$$
\left\{\begin{array}{l}
x=13, \quad 13 \\
y=27,33 \\
z=33, \quad 27
\end{array}\right.
$$