0 \le r < \infty, \qquad 0 \le \varphi < \frac{\pi}{2k},
\end{split}  \label{eq:TTW}$$ introduced by Tremblay, Turbiner and Winternitz (TTW) [@tremblay09], has aroused a lot of interest because of the conjectured superintegrability of the Hamiltonians for all positive integer values of $k$. For the corresponding classical systems, it has been shown that all bounded trajectories are closed and that the motion is periodic for all integer and rational values of $k$ [@tremblay10]. Such classical systems have been proved to be superintegrable [@kalnins09] and generalizable to higher dimensions [@kalnins10a]. For the quantum Hamiltonians, the validity of the superintegrability conjecture has been demonstrated for any odd integer $k$ [@cq10a] by using a $D_{2k}$ extension of $H_k$ [@cq10b] and a Dunkl operator formalism previously employed in the $k=3$ case [@cq95]. Furthermore, a canonical operator method has been applied to provide a constructive proof that all quantum Hamiltonians with rational $k$ are superintegrable [@kalnins10b].

Another attractive propery of the TTW family (\[eq:TTW\]) is that it includes as special cases several well-known Hamiltonians, which have been much studied in the literature both for their intrinsic mathematical properties and for their possible physical applications. They correspond to the Smorodinsky-Winternitz (SW) system ($k=1$) [@fris; @winternitz], the rational 