*(English)*Zbl 1037.34025

Authors’ abstract: Let $X(x,y)$ and $Y(x,y)$ be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system $\dot{x}=y+X(x,y),\dot{y}=Y(x,y)$, 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 $X=yf(x,{y}^{2})$ and $Y=g(x,{y}^{2})$, then the system has a local analytic first integral of the form $H={y}^{2}+F(x,y)$, 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 $H={y}^{2}+F(x,y)$, 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 $H={y}^{2}+F(x,y),$ where $F$ starts with terms of order higher than two.

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