Analytic integrability and characterization of centers for nilpotent singular points

Abstract

A method which provides necessary conditions to obtain a local analytic first integral in a neighborhood of a nilpotent singular point is developed. As an application we provide sufficient conditions in order that systems of the form % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaca % Gaeyypa0JaamyEaiabgUcaRiaadcfadaWgaaWcbaGaamOBaaqabaGc % caGGOaGaamiEaiaacYcacaWG5bGaaiykaiaacYcacaaMc8UabmyEay % aacaGaeyypa0JaamyuamaaBaaaleaacaWGUbaabeaakiaacIcacaWG % 4bGaaiilaiaadMhacaGGPaaaaa!4A27! $$\dot x = y + P_n (x,y),\,\dot y = Q_n (x,y)$$ where P n and Q n are homogeneous polynomials of degree n = 2, 3, 4, 5 have a local analytic first integral of the form H=y2+F(x, y), where F starts with terms of order higher than 2. We remark that, in general, the existence of such integral is only guaranteed when the singular point is a nilpotent center and the system has a formal first integral, see [6]. Therefore, we characterize the nilpotent centers of systems which have a local analytic first integral.