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 ẋ=x(1−a30x3−a21x2y−a12xy2−a03y3), ẏ=−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.