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In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.}{\bf 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 planeis superintegrable and that the additional integral of motion is a $2k$th-orderdifferential operator $Y_{2k}$. Here we demonstrate the conjecture for theinfinite family of Hamiltonians $H_k$ with odd $k \ge 3$, whose first membercorresponds to the three-body Calogero-Marchioro-Wolfes model after eliminationof the centre-of-mass motion. Our approach is based on the construction of some$D_{2k}$-extended and invariant Hamiltonian $\chh_k$, which can be interpretedas a modified boson oscillator Hamiltonian. The latter is then shown to possessa $D_{2k}$-invariant integral of motion $\cyy_{2k}$, from which $Y_{2k}$ can beobtained by projection in the $D_{2k}$ identity representation space.

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