On the Integrability of Two-Dimensional Flows

Analytic integrability of quadratic–linear polynomial differential systems

“…When Q(X, Y ) = X the finite singular point is (α, β) with α = 0, and when Q(X, Y ) = Y it is (α, β) with β = 0. So, doing the translation x = X − α, y = Y − β we obtain that system (6) becomes system (5), and the proposition is proved. Now we shall introduce some auxiliary results that will be used through the paper.…”

“…Quadratic polynomial differential systems have been investigated intensively, and more than one thousand papers have been published about these systems (see for instance [3,17,18,19]), but the problem of classifying all the integrable quadratic polynomial differential systems remains open. For more information on integrable differential systems in dimension 2, see for instance [5].…”

Polynomial and rational first integrals for planar homogeneous polynomial differential systems

“…To see the relation between the functional classes of the inverse integrating factors and their associated first integrals see Theorem 3 of [14], see also [7]. Many works deal with the integrability problem of a system of the form (2), see for instance [3,5,7,20,24] and references therein. The main idea of these works is to consider the singular points of the system and to give necessary conditions on the eigenvalues associated to the linear part of each singular point in order that the system has a first integral of a particular functional class.…”

Centers for a class of generalized quintic polynomial differential systems

“…the problem of distinguishing between a center and a focus. For singular points whose linear part has a pair of pure imaginary eigenvalues this problem is equivalent to the existence of an analytic first integral defined in a neighborhood of the singular point, see for more details the articles [24,25] and [2,13,14].…”