Local Integrability of Poincaré-Dulac Normal Forms

We consider dynamical systems in Poincaré-Dulac normal form having an equilibrium at the origin, and give a sufficient condition for them to be integrable, and prove that it is necessary for their special integrability under some condition. Moreover, we show that they are integrable if their resonance degrees are 0 or 1 and that they may be nonintegrable if their resonance degrees are greater than 1, as in Birkhoff normal forms for Hamiltonian systems. We demonstrate the theoretical results for a normal form appearing in the codimension-two fold-Hopf bifurcation.