Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems

Geometric Criterium in the Center Problem

“…F is orbitally equivalent to G = X h + ∞ j=1 a j h j (x, y)(x, y) T , see [3]. Choosing X = X h which has a center at the origin, we have…”

“…In the case b = 0, the origin of system is a center because the system is invariant by the symmetry (x, y, t) → (−x, y, −t). Following the ideas of [3], it is easy to prove that system (3.6) is equivalent tȯ…”

Nilpotent Systems Admitting an Algebraic Inverse Integrating Factor over $${{\mathbb{C}}((x,y))}$$

“…In the case, α We now prove that the condition is necessary. From Algaba et al [3], the system (4) is formally orbital equivalent to…”

“…These results extend previous results given in [6], [8,Theorem 5.2], where the authors consider elementary singularities that admit analytic orbital normalization. Algaba et al [3] characterize the nilpotent systems with a formal inverse integrating factor.…”

On the Formal Integrability Problem for Planar Differential Systems

“…However, there is no general method to provide the sufficiency for each family that satisfies some necessary conditions. The sufficiency is obtained verifying that the system is Hamiltonian, or that it has certain reversibility, or certain Lie symmetry, or finding a first integral well defined in a neighborhood of the singular point, sometimes finding an integrating factor that allows to construct a first integral, reducing the system to an integrable system; see, for instance, [4,5,[18][19][20][21][22][23] and the references therein. These methods have proved ineffective in certain families that already verify some necessary conditions, and in many papers some cases are established as open problems; see, for instance, [7,10,11].…”