Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems

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