The center problem for a 1:−4 resonant quadratic system

The main objective of this paper is to find necessary and sufficient conditions for a 1:−4 resonant system of the formx˙=x−a10x2−a01xy−a12y2,y˙=−4y+b2,−1x2+b10xy+b01y2 to have a center at the origin. Since applying a linear change of variables any system of this form can be transformed either to system with a10=1 or a10=0, these are the two cases considered here. When a10=1 there appear 46 resonant center conditions and for a10=0 there are 9 center conditions. To obtain necessary conditions for integrability the computation of the resonant saddle quantities (focus quantities) and the decomposition of the variety of the ideal generated by an initial string of them were used. The theory of Darboux first integrals and some other methods, as the monodromy arguments for instance, are used to show the sufficiency. Since decomposition of the variety mentioned above was performed using modular computations the obtained conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement.