Extensions of nonnatural Hamiltonians

The concept of extended Hamiltonian systems allows a geometric interpretation of several integrable and superintegrable systems with polynomial first integrals of a degree depending on a rational parameter. Until now, the extension procedure has been applied only in the case of natural Hamiltonians. We give several examples of application to nonnatural Hamiltonians, such as the Hamiltonian of a system of two point-vortices, the Hamiltonian of the Lotka–Volterra model, and some Hamiltonians quartic in the momenta. We effectively obtain extended Hamiltonians in some cases, fail in other cases, and briefly discuss the reasons for these results.