the solutions of (\[eq:riccati\]), we have been guided by the known results for Calogero-like Hamiltonians to be reviewed in section 5.

On insering (\[eq:F-C\]) in (\[eq:polar-bis\]), we obtain that for the even generators the operators $\Gamma$ and $Y$ become $$\begin{split}
  \Gamma & = \frac{k}{2\omega r^2}\biggl\{a \sec^2 k\varphi \biggl[\Bigl(\cos (k-2)\varphi \cos k\varphi
       + \frac{k}{2}(1 - \cos 2\varphi)\Bigr) {b^{\dagger}}_x b_x \\  
  & \quad - \Bigl(\sin (k-2)\varphi \cos k\varphi + \frac{k}{2} \sin 2\varphi\Bigr) ({b^{\dagger}}_x b_y + {b^{\dagger}}_y b_x) + 
       \Bigl(- \cos (k-2)\varphi \cos k\varphi \\
  & \quad + \frac{k}{2}(1 + \cos 2\varphi)\Bigr) {b^{\dagger}}_y b_y\biggr] + b \csc^2 k\varphi \biggl[\Bigl(\sin (k-2)
       \varphi \sin k\varphi + \frac{k}{2}(1 - \cos 2\varphi)\Bigr) {b^{\dagger}}_x b_x \\ 
  & \quad + \Bigl(\cos (k-2)\varphi \sin k\varphi - \frac{k}{2} \sin 2\varphi\Bigr) ({b^{\dagger}}_x b_y + {b^{\dagger}}_y b_x) +
       \Bigl(- \sin (k-2)\varphi \sin k\varphi \\
  & \quad + \frac{k}{2}(1 + \cos 2\varphi)\Bigr) {b^{\dagger}}_y b_y\biggr]\biggr\} 
\end{split}  \label{eq:Gamma}$$ and $$Y = \frac{1}{2} [{b^{\dagger}}_x b_x + {b^{\dagger}}_y b_y - k(a+b) - 1],  \label{eq:Y}$$ respectively. The supersymmetrized TTW Hamiltonian then assumes the form $${\cal H}^s = H_{k,{\rm B}} + H_{k,{\rm F}}, \qquad H_{k,{\rm B}} = H_k, \qquad H_{k,{\rm F}} = 4\omega
  (\Gamma + Y).  \label{eq:super-TTW}$$

Furthermore, the odd generators in (\[eq:polar\]) turn out to be $$\begin{split}
  V_{\pm} & 