The center problem. A view from the normal form theory

In this work, we analyze some aspects of the center problem from the perspective of the normal form theory. We provide alternative proofs of some well known results in the case of non-vanishing linear part (nondegenerate and nilpotent centers). Moreover, some new results are also derived. In particular, we show that the unique characterization for centers in this situation is the orbital reversibility. Also, we include some examples illustrating that this characterization is no longer valid for the case of null linear part.