AUSTRIAN-POLISH 2005 Problem 7

For each natural number $n\geq 2$ , solve the following system of equations in the integers $x_1, x_2, ..., x_n$ : $$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2, \forall 1\le i\le n$$ where $$S=x_1^2+x_2^2+\dots+x_n^2.$$