Nonintegrability of truncated Poincare-Dulac normal forms of resonance degree two

Abstract

We give sufficient conditions for three- or four-dimensional truncated Poincare-Dulac normal forms of resonance degree two to be meromorphically nonintegrable when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of simple planar systems, and use an approach for proving the meromorphic nonintegrability of planar systems, which is similar to but more sophisticated than the previously developed one. Our result also implies that general three- or four-dimensional systems are analytically nonintegrable if they are formally transformed into one of the truncated normal forms satisfying the sufficient conditions.