II. (20 points) Find all positive decimal integers $n$ that satisfy the following conditions: $n$ is an $a$-digit number, and $a^{a}=n$.


II. Since $1^{1}=1$, therefore, $n=1$ satisfies the condition.
When $a=2,3,4,5,6,7$, because
$$
\begin{array}{l}
2^{2}=4,3^{3}=27,4^{4}=256,5^{5}=3125, \\
6^{6}=46656,7^{7}=823543,
\end{array}
$$

thus, $n$ cannot be a two-digit number, three-digit number, $\cdots$, seven-digit number.
And when $a=8,9$,
$$
8^{8}=16777216,9^{9}=387420489,
$$

these two numbers satisfy the condition.
When $a \geqslant 10$, we have
$$
n=a^{a} \geqslant 10^{a}=1 \underbrace{00 \cdots 0}_{a \uparrow 0}>n,
$$

which is impossible.
Therefore, the positive integers that satisfy the condition are
$$
1,16777216,387420489 .
$$