Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems

摘要: We introduce a noncanonical ("new-time") transformation which exchanges the roles of coupling constant and energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable and, for example, explain observed duality between H\'enon-Heiles Holt models. It is shown that sometimes connect weak- full-Painlev\'e Hamiltonians. also discuss quantum integrability find origin deformation $\ensuremath{-}\frac{5}{72}{\ensuremath{\hbar}}^{2}{x}^{\ensuremath{-}2}$.