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Solution of the 1 : −2 resonant center problem in the quadratic case

Alexandra Fronville; Anton Sadovski; Henryk Żołądek — 1998

Fundamenta Mathematicae

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 $(x + A_{1} x^{2} + B_{1} x y + C y^{2}) \partial_{x} + (- 2 y + D x^{2} + A_{2} x y + B_{2} y^{2}) \partial_{y}$ . There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic...

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