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Local analytic integrability for nilpotent centers. (English) Zbl 1037.34025

Authors’ abstract: Let $X\left(x,y\right)$ and $Y\left(x,y\right)$ be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system $\stackrel{˙}{x}=y+X\left(x,y\right),\stackrel{˙}{y}=Y\left(x,y\right)$, has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following:

(1) If $X=yf\left(x,{y}^{2}\right)$ and $Y=g\left(x,{y}^{2}\right)$, then the system has a local analytic first integral of the form $H={y}^{2}+F\left(x,y\right)$, where $F$ starts with terms of order higher than two.

(2) If the system has a formal first integral, then it has a formal first integral of the form $H={y}^{2}+F\left(x,y\right)$, where $F$ starts with terms of order higher than two. In particular, if the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form $H={y}^{2}+F\left(x,y\right),$ where $F$ starts with terms of order higher than two.

As an application, we characterize the nilpotent centers for the differential systems (3) $\stackrel{˙}{x}=y+{P}_{3}\left(x,y\right)$, $\stackrel{˙}{y}={Q}_{3}\left(x,y\right)$, which have a local analytic first integral, where ${P}_{3}$ and ${Q}_{3}$ are homogeneous polynomials of degree three.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C20 Transformation and reduction of ODE and systems, normal forms 34C25 Periodic solutions of ODE
##### Keywords:
planar vector field; nilpotent center