  Codimension-two bifurcations are fundamental and interesting phenomena in
dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most
important among them. We study the unfoldings of these two codimension-two
bifurcations, and obtain sufficient conditions for their nonintegrability in
the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to
those of planar polynomial vector fields and analyze the nonintegrability of
the planar vector fields, based on Ayoul and Zung's version of the
Morales-Ramis theory. New useful criteria for nonintegrability of planar
polynomial vector fields are also given. The approaches used here are
applicable to many problems including circular symmetric systems.
