  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.
