On the kolmogorov system with fourth-degree homogeneous nonlinearities

The necessary conditions of the center at the singular point O (0, 0) for the kolmogorov system with fourth-degree homogeneous nonlinearities are found in the work of B. Ferenc, J. Dzheny, W. Liu, V. G. Romanovsky. For all these cases, with the exception of two, sufficiency is also proved. In the present article, sufficiency is proved for two cases of the center that were not proved in the above-mentioned work. In addition, the sufficiency of two other conditions of the center is proved in another way.