Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd k

In a recent communication paper by Tremblay et al (2009 J. Phys. A: Math. Theor. 42 205206), it has been conjectured that for any integer value of k, some novel exactly solvable and integrable quantum Hamiltonian H(k) on a plane is superintegrable and that the additional integral of motion is a 2kth- order differential operator Y(2k). Here we demonstrate the conjecture for the infinite family of Hamiltonians H(k) with odd k >= 3, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some D(2k)-extended and invariant Hamiltonian H(k), which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a D(2k)-invariant integral of motion Y(2k), from which Y(2k) can be obtained by projection in the D(2k) identity representation space.