Authors’ abstract: “The 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
There are twenty cases of center. Their necessity is proved by A. Fronville [Algorithmic approach to the center problem for 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”.