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Solution of the 1: -2 resonant center problem in the quadratic case. (English) Zbl 0943.34018

Authors’ abstract: “The $1:-2$ resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field

$\left(x+{A}_{1}{x}^{2}+{B}_{1}xy+C{y}^{2}\right){\partial }_{x}+\left(-2y+D{x}^{2}+{A}_{2}xy+{B}_{2}{y}^{2}\right){\partial }_{y}·$

There are twenty cases of center. Their necessity is proved by A. Fronville [Algorithmic approach to the center problem for $1:-2$ resonant singular points of polynomial vector fields, Nonlinearity, submitted; not yet available for Zbl ] using factorization of polynomials with integer coefficients modulo prime numbers. Here, it is shown that, in each of the twenty cases, there is an analytic first integral. The authors develop a new method of investigation of analytic properties of polynomial vector fields”.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 37C20 Generic properties, structural stability