Analytic integrability and characterization of centers for nilpotent singular points (Q1766753)

The author studies the existence of local analytic first integrals for nilpotent singular points, i.e., for the singular point at the origin of the following system \[ \dot{x}=y+X(x,y),\quad \dot{y}=Y(x,y), \] where \(X(x,y)\) and \(Y(x,y)\) are analytic functions without linear and constant terms in a certain neighborhood of the origin. The paper contains a method which provides necessary conditions to obtain such an integral in a neighborhood of these singular points. The author also provides sufficient conditions in case that \(X(x,y)\) and \(Y(x,y)\) are homogeneous polynomials of degree \(n=2,3,4,5\).