= \frac{1}{2\sqrt{\omega}} \biggl[\biggl(\mp \cos \varphi \partial_r  \pm \frac{1}{r} \sin \varphi
        \partial_{\varphi} + \omega r \cos \varphi \pm \frac{ka}{r} \frac{\cos (k-1)\varphi}{\cos k\varphi} \\
  & \quad \pm \frac{kb}{r} \frac{\sin (k-1)\varphi}{\sin k\varphi}\biggr) {b^{\dagger}}_x + \biggl(\mp \sin \varphi 
        \partial_r \mp \frac{1}{r} \cos \varphi \partial_{\varphi} + \omega r \sin \varphi \\
  & \quad \mp \frac{ka}{r} \frac{\sin (k-1)\varphi}{\cos k\varphi} \pm \frac{kb}{r} \frac{\cos (k-1)\varphi} 
        {\sin k\varphi}\biggr) {b^{\dagger}}_y \biggr], \\
  W_{\pm} & = \frac{1}{2\sqrt{\omega}} \biggl[\biggl(\mp \cos \varphi \partial_r  \pm \frac{1}{r} \sin \varphi
        \partial_{\varphi} + \omega r \cos \varphi \mp \frac{ka}{r} \frac{\cos (k-1)\varphi}{\cos k\varphi} \\
  & \quad \mp \frac{kb}{r} \frac{\sin (k-1)\varphi}{\sin k\varphi}\biggr) b_x + \biggl(\mp \sin \varphi 
        \partial_r \mp \frac{1}{r} \cos \varphi \partial_{\varphi} + \omega r \sin \varphi \\
  & \quad \pm \frac{ka}{r} \frac{\sin (k-1)\varphi}{\cos k\varphi} \mp \frac{kb}{r} \frac{\cos (k-1)\varphi} 
        {\sin k\varphi}\biggr) b_y \biggr], 
\end{split}  \label{eq:V-W}$$ with $W_+$ and $V_-$ providing the two supercharge operators for the supersymmetrized Hamiltonian (\[eq:super-TTW\]) through equation (\[eq:SUSY\]).

By introducing two ’rotated’ pairs of fermionic creation and annihilation operators $({\bar{b}^{\dagger}}_x, {\bar{b}}_x)$ and $({\bar{b}^{\dagger}}_y, {\bar{b}}_y)$, defined by $${\bar{b}^{\dagger}}_x = {b^{\dagger}}_x \cos \varphi + {b^{\dagger}}_y \sin \varphi, \qquad {\bar{b}^{\dagger}}_y = - {b^{\dagger}}_x \sin \varphi + {b^{\dagger}}_y \cos \varphi$$ and similarly for ${\bar{b}}_x$, ${\bar{b}}_y$, equations (\[eq:Gamma\]), (\[eq:Y\]) and (\[eq:V-W\]) can be recast in a somewhat simpler form[^1] 