This paper concludes a series of reviews on the results of scientific research conducted at the Department of Differential Equations of St. Petersburg State University over the past three decades. This study highlights the results obtained by Professor A.F. Andreev and his students in the field of the local qualitative theory of differential equations. The first part of the paper presents the results of Andreev and his co-authors on the center–focus problem in two-dimensional systems of differential equations with a nilpotent linear part and nonlinearities that are homogeneous polynomials of odd degree. Necessary and sufficient coefficient conditions for the presence of a center in such systems are indicated and the question of the existence of limit cycles in the neighborhood of the origin of coordinates is explored when the nonlinearities are seventh-degree polynomials. The second part of the work is devoted to research by A.F. Andreev and I.A. Andreeva on one family of autonomous two-dimensional differential systems, the right-hand sides of which are mutually simple homogeneous third- and second-degree polynomials. Complete classification of the global phase portraits of systems of the specified family in the Poincaré disk is given, and a proof of the absence of limit cycles in systems of this type is obtained.
Andreeva et al. (Sun,) studied this question.