Integrability conditions for Lotka–Volterra planar complex quintic systems

In this paper we obtain necessary and sufficient integrability conditions at the origin for the Lotka–Volterra complex quintic systems which are linear systems perturbed by fifth degree homogeneous polynomials, i.e., we consider systems of the form ẋ=x(1−a40x4−a31x3y−a22x2y2−a13xy3−a04y4), ẏ=−y(1−b40x4−b31x3y−b22x2y2−b13xy3−b04y4). The necessity of these conditions is derived from the first nine focus-saddle quantities and their sufficiency is proved by finding an inverse integrating factor or a first integral.