The Tremblay-Turbiner-Winternitz system as extended Hamiltonian

We generalize the idea of "extension of Hamiltonian systems" -- developed in a series of previous articles -- which allows the explicit construction of Hamiltonian systems with additional non-trivial polynomial first integrals of arbitrarily high degree, as well as the determination of new superintegrable systems from old ones. The present generalization, that we call "modified extension of Hamiltonian systems", produces the third independent first integral for the (complete) Tremblay-Turbiner-Winternitz (TTW) system, as well as for the caged anisotropic oscillator in dimension two.