Analytically integrable centers of perturbations of cubic homogeneous systems (Q2037343)

The authors investigate analytic integrability and homogenization problem of planar polynomial differential systems, whose homogeneous component is a cubic homogeneous vector field. As a first step, they give complete norm forms of the cubic homogeneous vector fields having a polynomial first integral, using the results and methods in [\textit{A. Algaba} et al., Rocky Mt. J. Math. 41, No. 1, 1--22 (2011; Zbl 1213.37093)], which is a necessary condition for the analytic integrability of their considered perturbations of cubic homogeneous systems. Secondly, they use the notion of Lie symmetry to prove the vector field is analytically integrable if and only if it is orbitally equivalent to its cubic homogeneous component. They also point out that the analytic integrability of the considered vector field can be solved via the formal inverse integrating factor. Some examples are given to illustrate the results.