Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post–Winternitz System

The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient conditions are found in order that the transformed high-degree first integral of the transformed Hamiltonian is determined by the same algorithm which computes the corresponding first integral of the original extended Hamiltonian. As examples, we consider the Post–Winternitz system and the 2D caged anisotropic oscillator.