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

The Tremblay-Turbiner-Winternitz system as extended Hamiltonian