Linearizability conditions for Lotka–Volterra planar complex quartic systems having homogeneous nonlinearities

In this paper we investigate the linearizability problem for the two-dimensional Lotka–Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, i.e., we consider systems of the form ˙x = x(1−a30x3 − a21x2y − a12xy2 − a03y3), ˙y = −y(1 − b30x3 − b21x2y − b12xy2 − b03y3). The necessary and sufficient conditions for the linearizability of this system are found. From them the conditions for isochronicity of the corresponding real system can be derived.