  The concept of extended Hamiltonian systems allows the geometrical
interpretation of several integrable and superintegrable systems with
polynomial first integrals of degree depending on a rational parameter. Until
now, the procedure of extension has been applied only in the case of natural
Hamiltonians. In this article, we give several examples of application to
non-natural Hamiltonians, such as the two point-vortices, the Lotka-Volterra
and some quartic in the momenta Hamiltonians, obtaining effectively extended
Hamiltonians in some cases and failing in others. We briefly discuss the
reasons of these results.
