Singularities of integrable and near-integrable Hamiltonian systems

The aim of this Letter is to show that the singularities of integrable Hamiltonian systems, besides being important for such systems themselves, also have many applications in the study of near-integrable systems. In particular, we will show how they are related to Kolmogorov's nondegeneracy condition (in the famous KAM theorem), the Poincare-Melnikov function and its generalizations, topological entropy, and nonintegrability criteria.