  We prove the integrability and superintegrability of a family of natural
Hamiltonians which includes and generalises those studied in some literature,
originally defined on the 2D Minkowski space. Some of the new Hamiltonians are
a perfect analogy of the well-known superintegrable system on the Euclidean
plane proposed by Tremblay-Turbiner-Winternitz and they are defined on
Minkowski space, as well as on all other 2D manifolds of constant curvature,
Riemannian or pseudo-Riemannian. We show also how the application of the
coupling-constant-metamorphosis technique allows us to obtain new
superintegrable Hamiltonians from the previous ones. Moreover, for the
Minkowski case, we show the quantum superintegrability of the corresponding
quantum Hamiltonian operator.Our results are obtained by applying the theory of
extended Hamiltonian systems, which is strictly connected with the geometry of
warped manifolds.
