This problem was first asked at Mathematics Stack Exchange, where it wasn't paid much attention.

Define an equation $A_n$ like the following: $$\sum^{n}_{i=1}\sum^{n}_{j=1,j\neq i}(x_{i})^{(x_j)}=(\sum^{n}_{i=1}x_i)^{(\sum^{n}_{i=1}x_i)}$$ For example, $A_3$ looks like the following equation: $${x_1}^{x_2}+{x_1}^{x_3}+{x_2}^{x_1}+{x_2}^{x_3}+{x_3}^{x_1}+{x_3}^{x_2}=({x_1}+{x_2}+{x_3})^{({x_1}+{x_2}+{x_3})}$$

Let's assume that for every non-negative integer solutions for $A_n$, $x_i\leq x_{i+1}$ for every $1\leq i<n$, then there are two distinct non-negetive solutions for $A_3$, one is ${x_1}=0,{x_2}=0,{x_3}=2$, and the other is ${x_1}=0,{x_2}=1,{x_3}=1$.

We call a solution for $A_n$ 'non-trivial' if $x_{n-1}\neq0$. The only known non-trivial solution is ${x_1}=0,{x_2}=1,{x_3}=1$ for $A_3$. The problem is: are there any more non-trivial solutions for $A_n$?
If so, please give an example.

Since this question is difficult enough, I will also recieve answers which give some features about every non-trivial solutions.