Integrability Conditions for Lotka-Volterra Planar Complex Quartic Systems Having Homogeneous Nonlinearities

In this paper we investigate the integrability problem for the two-dimensional Lotka-Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, that is, we consider systems of the form $\dot{x}=x(1-a_{30}x^{3}-a_{21} x^{2} y-a_{12}x y^{2} -a_{03}y^{3})$ , $\dot{y}=-y(1-b_{30}x^{3}-b_{21} x^{2} y-b_{12}x y^{2}-b_{03} y^{3})$ . Conditions for the integrability of this system are found. From them the center conditions for corresponding real system can be derived. The study relays on making use of algorithms of computational algebra based on the Groebner basis theory. To simplify laborious manipulations with polynomial modular arithmetics is involved.