The paper is a study of the integrability problem for the planar systems , where , and is a formal or analytic vector-function that vanishes at the origin and can be expressed as a perturbation of a generic Hamiltonian quasi-homogeneous vector-function with higher-degree quasi-homogeneous terms (such type of functions is naturally connected with Newton polyhedra and often used in the analysis of singularities. The authors consider the formal integrability problem for such systems, but in some analytical cases the convergence of the corresponding formal series can be established, and the formal integrability implies the analytical one.
The main result of the paper is that the quoted system with some additional conditions is formally integrable if and only if it is formally conjugated, via dissipative transformations, to a divergence-free system. To derive this result, the authors obtain a suitable quasi-homogeneous -smooth normal form for such systems. As an application they compute the above quasi-homogeneous normal form for various concrete systems, for instance, for the Bogdanov–Takens singularities of different types (nilpotent linear part) and for a family of planar systems with zero linear part, and obtain the explicit conditions of the formal integrability for these systems.