The solution of the 1:−3 resonant center problem in the quadratic case

The 1:−3 resonant center problem in the quadratic case is to find necessary and sufficient conditions for the existence of a local analytic first integral for the differential systemẋ=x-a10x2-a01xy-a12y2,ẏ=-3y+b21x2+b10xy+b01y2.There appear 25 center cases for a10=1a10=1 and 11 cases for a01=0a01=0. The necessity is obtained using modular arithmetics and with high probability we have all the resonant center cases. We show that in each case there exists a local analytic first integral (sufficient condition) around the origin. This sufficient condition is proved using different classical criteria and in one case monodromy arguments.