Periodic orbits and non-existence of \(C^1\) first integrals for analytic differential systems exhibiting a zero-Hopf bifurcation in \(\mathbb {R}^4\)

In this paper we investigate a particular case of a zero-Hopf bifurcation of a four dimensional analytic differential system. We prove that at most five periodic orbits bifurcate from the zero-Hopf equilibrium using the averaging theory of first order and give a specific example to illustrate this conclusion. Moreover we prove the non-existence of \(C^1\) $C^1$ first integrals in a neighbourhood of these periodic orbits.