$$\begin{split}
  {\cal H}^s_{\rm rel} & = - \partial_r^2 - \frac{1}{r} \partial_r - \frac{1}{r^2} \partial_{\varphi}^2 + \omega^2
        r^2 + \frac{1}{r^2 \cos^2 \varphi} a (a + [{b^{\dagger}}_x, b_x]) \\
  & \quad + \frac{1}{r^2 \cos^2 \left(\varphi - \frac{2\pi}{3}\right)} a \left(a + \frac{1}{4} \left[\sqrt{3} {b^{\dagger}}_y
        - {b^{\dagger}}_x, \sqrt{3} b_y - b_x\right]\right) \\
  & \quad + \frac{1}{r^2 \cos^2 \left(\varphi - \frac{4\pi}{3}\right)} a \left(a + \frac{1}{4} \left[\sqrt{3} {b^{\dagger}}_y
        + {b^{\dagger}}_x, \sqrt{3} b_y + b_x\right]\right) \\
  & \quad + \frac{1}{r^2 \sin^2 \varphi} b (b + [{b^{\dagger}}_y, b_y]) \\
  & \quad + \frac{1}{r^2 \sin^2 \left(\varphi - \frac{2\pi}{3}\right)} b \left(b + \frac{1}{4} \left[\sqrt{3} {b^{\dagger}}_x
        + {b^{\dagger}}_y, \sqrt{3} b_x + b_y\right]\right) \\
  & \quad + \frac{1}{r^2 \sin^2 \left(\varphi - \frac{4\pi}{3}\right)} b \left(b + \frac{1}{4} \left[\sqrt{3} {b^{\dagger}}_x
        - {b^{\dagger}}_y, \sqrt{3} b_x - b_y\right]\right) \\
  & \quad + 2\omega ({b^{\dagger}}_x b_x + {b^{\dagger}}_y b_y) - 2\omega (3a+3b+1),  
\end{split}  \label{eq:super-rel}$$ $$\begin{split}
  Q_{\rm rel} & = \left(- \cos \varphi \partial_r + \frac{1}{r} \sin \varphi \partial_{\varphi} + \omega r 
        \cos \varphi - \frac{3a}{r} \frac{\cos 2\varphi}{\cos 3\varphi} - \frac{3b}{r} 
        \frac{\sin 2\varphi}{\sin 3\varphi}\right) b_x \\ 
  & \quad + \left(- \sin \varphi \partial_r - \frac{1}{r} \cos \varphi \partial_{\varphi} + \omega r 
        \sin \varphi + \frac{3a}{r} \frac{\sin 2\varphi}{\cos 3\varphi} - \frac{3b}{r} 
        \frac{\cos 2\varphi}{\sin 3\varphi}\right) b_y
\end{split}  \label{eq:Q-rel}$$ and $$\begin{split}
  