Example: 15 Given positive integers $x, y, z$ satisfy $x^{3}-y^{3}-z^{3}=3 x y z, x^{2}=2(y+z)$. Find the value of $x y+y z+z x$.

---

The translation is provided as requested, maintaining the original text's line breaks and format.
$$
\begin{array}{l}
\text { Solution: } x^{3}-y^{3}-z^{3}-3 x y z \\
=x^{3}-(y+z)^{3}-3 x y z+3 y^{2} z+3 y z^{2} \\
=(x-y-z)\left(x^{2}+x y+x z+y^{2}+2 y z\right. \\
\left.+z^{2}\right)-3 y z(x-y-z) \\
=(x-y-z)\left(x^{2}+y^{2}+z^{2}+x y-y z+x z\right) \\
=\frac{1}{2}(x-y-z)\left[(x+y)^{2}+(y-z)^{2}\right. \\
\left.+(z-x)^{2}\right] \text {. } \\
\end{array}
$$

From the given, we have $x-y-z=0$,
$$
\therefore x=y+z \text {. }
$$

Substituting $x^{2}=2(y+z)$, we get $x=2, y+z=2$. Given $y \geqslant 1, z \geqslant 1$, we have $y=z=1$.
$$
\therefore x y+y z+z x=5 \text {. }
$$