2. Solve the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n}=n \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=n \\
\cdots \cdots \\
x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=n
\end{array}\right.
$$
( Hint: Using Newton's formulas and the initial conditions $S_{1}=$ $S_{2}=\cdots=S_{n}=n$, we get $\sigma_{1}-\sigma_{2}+\cdots+(-1)^{n-2} \sigma_{n-1}+(-1)^{n-1} \sigma_{n}=1$. Therefore, among $x_{1}, x_{2}, \cdots, x_{n}$, there must be one that is 1. By symmetry, assume $x_{n}=1$, which reduces the original system of equations. Then, repeat the above operation on the reduced system of the first $n-1$ equations, yielding $x_{n-1}=1$. Continue simplifying the system of equations, repeating this process, and after a finite number of operations, we obtain the unique solution $x_{1}=x_{2}=\cdots=x_{n}=1$. )