The 1: −q resonant center problem for certain cubic Lotka–Volterra systems

Necessary conditions and distinct sufficient conditions are derived for the system x˙=x(1-a20x2-a11xy-a02y2), y˙=y(-q+b20x2+b11xy+b02y2) to admit a first integral of the form Φ(x,y)=xqy+⋯Φ(x,y)=xqy+⋯ in a neighborhood of the origin, in which case the origin is termed a 1:-q1:-q resonant center. Necessary and sufficient conditions are obtained for odd q,q⩽9q,q⩽9; necessary conditions, most of which are also sufficient, are obtained for even q,q⩽8q,q⩽8. Key ideas in the proofs are computation of focus quantities for the complexified systems and decomposition of the variety of the ideal generated by an initial string of them to obtain necessary conditions, and the theory of Darboux first integrals to show sufficiency.