From the equation we know that $S(n)$ must be itself a perfect square, and since $n<1000$, $1 \leq S(n) \leq 27$. Thus the only possible values for $S(n)$ are $1, 4, 9, 16$ and $25$. Moreover, \[n \equiv S(n) (mod 9) \rightarrow n^2 \equiv n^3 (mod 9) \rightarrow n^2(n-1) \equiv 0 (mod 9)\]but $\gcd(n^2,n-1)=1$ so $9|n^2$ or $9|n-1$, leading to $S(n)=1$ or $S(n)=9$, substituting in the equations we see that the only solutions are $1$ and $27$.