  We prove that general three- or four-dimensional systems %of differential
equations are real-analytically nonintegrable near degenerate equilibria in the
Bogoyavlenskij sense under additional weak conditions 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 the corresponding
Poincare-Dulac normal forms and further to that of simple planar systems, and
use a novel approach for proving the analytic nonintegrability of planar
systems. Our result also implies that general three- and four-dimensional
systems exhibiting fold-Hopf and double-Hopf codimension-two bifurcations,
respectively, are real-analytically nonintegrable under the weak conditions. To
demonstrate these results, we give two examples for the Rossler system and
coupled van der Pol oscillators.
