observed in other contexts [@ghosh04; @hakobyan].

The $k=1$, 2 and 3 cases
========================

The purpose of this section is to establish some connections between the outcomes of section 3 and some known results corresponding to $k=1$, 2 and 3. For simplicity’s sake, we will restrict ourselves here to the explicit relations obtained for ${\cal H}^s$ and $Q$, defined in (\[eq:SUSY\]), as the remaining ${\rm osp}(2/2, {{\mathbb{R}}})$ generators can be easily dealt with in the same way.

SW model ($k=1$)
----------------

The SW Hamiltonian [@fris; @winternitz] $$H_1 = - \partial_x^2 - \partial_y^2 + \omega^2 (x^2 + y^2) + \frac{a(a-1)}{x^2} + \frac{b(b-1)}{y^2}$$ is usually written in cartesian coordinates, wherein it is also separable.

On setting $k=1$ in (\[eq:Gamma\]), (\[eq:Y\]), (\[eq:super-TTW\]) and (\[eq:V-W\]) and going back from $r$, $\varphi$ to $x$, $y$, we directly get $$\begin{split}
  {\cal H}^s & = - \partial_x^2 - \partial_y^2 + \omega^2 (x^2 + y^2) + \frac{a^2}{x^2} + \frac{b^2}{y^2}
        + 2\omega ({b^{\dagger}}_x b_x + {b^{\dagger}}_y b_y) + \frac{a}{x^2} [{b^{\dagger}}_x, b_x] \\
  & \quad + \frac{b}{y^2} [{b^{\dagger}}_y, b_y] - 2\omega (a+b+1), \\
  Q & = \left(- \partial_x + \omega x - \frac{a}{x}\right) b_x + \left(- \partial_y + \omega y - \frac{b}{y}\right)
        b_y. 
\end{split}$$ Similar results would be obtained either by 