Families of Hamiltonians sharing a common symmetry structure

We describe a general procedure which allows to construct, given a family of Hamiltonians H([under q]̲,[under p]̲;[under λ]̲), a single new Hamiltonian H[over ̃]([under q]̲,[under p]̲) such that, on each level set of H[over ̃], the unparametrized orbits are those of a level set of H with the values of parameters [under λ]̲ varying with the value of H[over ̃]. The symmetry structure of H[over ̃] can be completely characterized, provided the symmetries of initial family are known. Our approach covers numerous models considered in literature as well as it allows to construct novel ones. It provides a far reaching generalization of the Hietarinta et al. coupling-constant metamorphosis method and another proof of the Darboux theorem.