Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$

Global bifurcations in the generic one-parameter families that unfold a vector field with a separatrix loop on the two-sphere are described. The sequence of bifurcations that occurs is in a sense in one-to-one correspondence with finite sets on a circle having some additional structure on them. Families under study appear to be structurally stable. The main tool is the Leontovich–Mayer–Fedorov (LMF) graph, analog of the separatrix sceleton and an invariant of the orbital topological classification of the vector fields on the two-sphere. Its properties and applications are described.