Integrability and generalized center problem of resonant singular point

Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor

“…If λ 1 /λ 2 = −p/q ∈ Q then we have a resonant saddle. A resonant saddle has an analytic first integral around the singular point if, and only if, it is orbitally linearizable, see for instance [17,22,28] and references therein. The analytic integrability problem for a nilpotent singularity has been recently theoretically characterized in [11,Theorem 1.2], see also below.…”

On a conjecture on the integrability of Liénard systems

“…The study of the existence or not of a first integral in a neighborhood of a [p : −q] resonant saddle is a difficult problem, see for instance [2,7,8,12,14] and the references quoted there. Hence Theorem 2 says that the study of the existence or not of a first integral in a neighborhood of a strong saddle for the Liénard differential system (2) is also difficult.…”

“…From now on we assume that k 1 > k 2 because the other case is done in a similar manner. Hence we take (12) z = u k 2 v k 1 that is v = z 1/k 1 u −k 2 /k 1 .…”

Integrability of Liénard systems with a weak saddle

“…, and the so-called [p : −q] resonant saddle quantities v i are polynomials of the coefficients of system (2). If all the v i are zero we say that we have a formal analytic resonant saddle, see [11,26] and references therein. From this result we obtain also the existence of a local analytic first integral, see [21].…”