(本题满分 9 分) 设有齐次线性方程组 $$ \left\{ \begin{array}{l} (1 + a) x _ {1} + x _ {2} + \dots + x _ {n} = 0, \\ 2 x _ {1} + (2 + a) x _ {2} + \dots + 2 x _ {n} = 0, \\ \dots \dots \\ n x _ {1} + n x _ {2} + \dots + (n + a) x _ {n} = 0, \end{array} \right. \quad (n \geqslant 2), $$ 试问 $a$ 取何值时，该方程组有非零解，并求出其通解