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 $$y_1+\ldots+y_{n}=y_{n+1}+\ldots y_{2n}.$$ With non trivial I mean the ones like $$y_1=y_{n+1},\ldots,y_{n}=y_{2n},$$ or their permutations.

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.