\(n-1\) independent first integrals for linear differential systems in \(\mathbb{R}^n\) or \(\mathbb{C}^n\) (Q1773858)

For the linear differential system \[ \dot{x}=Ax,\;x\in \mathbb{R}^n (\mathbb{C}^n), \tag{1} \] with constant matrix \(A\), the authors construct \((n-1)\) independent Darboux first integrals. It follows that any linear system (1) with constant coefficients is Darboux integrable.