Example 1 Find all positive integer solutions of the system of equations
$$
\left\{\begin{array}{l}
a^{3}-b^{3}-c^{3}=3 a b c, \\
a^{2}=2(b+c)
\end{array}\right.
$$
From equation (1) we get
$$
(a-b-c)\left[a^{2}+(b-c)^{2}+a b+b c+c a\right]=0 \text {. }
$$

Since the second factor on the left side cannot be zero, then
$$
a=b+c=\frac{1}{2} a^{2} .
$$

Thus, the unique positive integer solution to the system of equations is
$$
(a, b, c)=(2,1,1) .
$$