The question is very simple and I apologize for that, but I am not an expert of this kind of problem.
Given the polynomial
$$ P(x\_1,\ldots,x\_{2n})=x\_1^2+\ldots+x\_n^2-x\_{n+1}^2-\ldots-x\_{2n}^2,$$
I would like to know if there are non trivial integer roots $(y\_1,\ldots, y\_{2n})$ such that
$$y\_1+\cdots+y\_{n}=y\_{n+1}+\cdots+ y\_{2n}.$$
With non trivial I mean the ones like
$$y\_1=y\_{n+1},\ldots,y\_{n}=y\_{2n},$$
or their permutations.



Integer roots of a symmetric polynomial
Fix large $N$ and consider all $n$-tuples $(x\_1,\dots,x\_n)\in \{1,\dots,N\}^n$. There are $N^n$ such $n$-tuples, at least $N^n/n!$ tuples modulo permutations, and for them the pairs $(x\_1+\dots+x\_n,x\_1^2+\dots+x\_n^2)$ take at most $n\cdot N\cdot n\cdot N^2=n^2N^3$ possible values. Thus by pigeonhole principle some value is obtained at least $N^{n-3}/(n^2\cdot n!)$ times. This is greater than 1 if $n>3$ and $N$ is chosen large enough.


