Non-degenerate singularities of integrable dynamical systems

We give a natural notion of non-degeneracy for singular points of integrable non-Hamiltonian systems, and show that such non-degenerate singularities are locally geometrically linearizable and deformation rigid in the analytic case. We conjecture that the same result also holds in the smooth case, and prove this conjecture for systems of type $(n, 0)$, i.e. $n$ commuting smooth vector fields on an $n$-manifold.