Analytic integrability around a nilpotent singularity: The non-generic case

Abstract

Recently, in [9] is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. In this work we solve the remaining case completing the analytic integrability problem for such singularity.

Algebraic integrability of nilpotent planar vector fields

“…for some integer r, where F j = (P j+ t 1 , Q j+ t 2 ) T ∈ Q t j and F r ≡ 0 . As mentioned before, the nilpotent vector fields are analytically integrable if, and only if, they are orbitally equivalent to a polynomially integrable quasi-homogeneous vector fields, see [1][2][3] . In this paper, we solve the remaining case, completing the algebraic integrability problem for such singularity; that is, we address our study when the vector field has an algebraic first integral but it does not have any analytic first integral and we obtain the following result.…”

“…Let prove the necessary condition. From Proposition 5 , after a polynomial change of variables (if necessary) is transformed into (2) .…”

“…If the vector field F is analytically integrable then there exists , a polynomial change of variables, such that the lowest-degree quasihomogeneous term * F := F r is polynomially integrable. We distinguish three cases: for F r Hamiltonian vector field, we apply [1, Theorem 3.19] ; for F r non-Hamiltonian and irreducible, we apply [3, Theorem 1.2] ; and for F r non-Hamiltonian and reducible we apply [2,Theorem 1.4] . In all the cases, the analytic integrability leads to F orbitally equivalent to F r .…”

“…Recently, Algaba et al [3] have solved the analytic integrability problem around a nilpotent singularity of a planar vector field under generic conditions (the origin of the leader quasi-homogeneous vector field is an isolated singular point), see [3,Theorem 1.2] . Later, Algaba et al [2] have solved the remaining case, completing the analytic integrability problem for such singularity, see [2,Theorem 1.4] . In all the cases, the vector field has an analytic local first integral if, and only if, it is orbitally quasi-homogenizable (orbitally equivalent to its quasihomogeneous leader term).…”