Analytic integrability around a nilpotent singularity

Abstract

In this work it is solved the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions.

Orbital Reversibility of Planar Vector Fields

“…Notice that P t k−r = {0} if k < r and P t 0 = span{1}. Next result is an adaptation to our case of [40] [Lemma 3.6].…”

“…So r+k+|t| (g) ∈ h . As h has simple factor h = ∏ s i=1 f i , where each f i is a simple factor, therefore X h is irreducible and r+k+|t| (g) ∈ f i for all 1 ≤ i ≤ s. By [40] [Lemma 3.21] we get g ∈ f i and then g ∈ h ∩ ∆ k'+|t| which is a contradiction. Proof.…”

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

“…The next theorem is the main result of [9] that solves the case B) of Proposition 1.1, see also [10] where the characterization is through the existence of an inverse integrating factor.…”

A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

“…The next result establishes the normal preform of the quasi-homogeneous leading term. Proposition 2.3 [8,Proposition 2.5]. Considerẋ = F(x) a nilpotent vector field with 0 an isolated singular point.…”

“…Lemma 2.8 [8,Lemma 3.24]. Consider p ∈ P t k , F n = X h + μD 0 ∈ Q t n irreducible and f ∈ P t s an irreducible quasi-homogeneous invariant curve of F n with K n its cofactor.…”