Center configurations of Hamiltonian cubic systems

Second order eyes of Hamiltonian cubic systems are classified into seven classes based on the orientation of the cycles within these eyes. They are further categorized into nine Conti classes based on the structure of the separatrix cycles that bound them. Examples of systems of each type are presented, and examples of third and fourth order eyes are also given. The classification is connected to part of Hilbert's sixteenth problem which asks for the possible relative positions of cycles in an autonomous polynomial system in the plane.