Nonexistence of formal first integrals for general nonlinear systems under resonance

In this paper, we consider the nonintegrability for nonlinear systems under the simple resonant case, i.e., the Jacobian matrix of vector field at some fixed point has some single multiply zero eigenvalues, and some nonzero eigenvalues which are N-independent. By using the Poincaré–Dulac normal form theory, we give a necessary condition for the system under consideration to have formal first integral.