In this paper we present hypernormal forms up to an arbitrary order for equilibria of tridimensional systems having a linear degneracy corresponding to a pair of pure imaginary eigenvalues and a third one zero. These simplest normal forms are obtained assuming some generic conditions on the quadratic terms, and using C∞-conjugacy as well as C∞-equivalence. Also, the case of ℤ2-symmetric systems is considered. In this situation, the hypernormal forms are characterized under generic conditions on the cubic terms. In all the cases, we provide recursive algorithms that compute explicitly the hypernormal form coefficients, in terms of the normal form coefficients.

In this paper we consider perturbations of quasi-homogeneous planar Hamiltonian systems, where the Hamiltonian function does not contain multiple factors. It is important to note that the most interesting cases (linear saddle, linear centre, nilpotent case, etc) fall into this category. For such kinds of systems, we characterize the integrability problem, by connecting it with the normal form theory.

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