case where there are only two variables $x_1$, $x_2$, which are the cartesian coordinates $x$, $y$ of a particle on a plane, and consequently two pairs of fermionic creation and annihilation operators, denoted by $({b^{\dagger}}_x, b_x)$ and $({b^{\dagger}}_y, b_y)$. On setting $x = r \cos \varphi$, $y = r \sin \varphi$, $W$ becomes a function of $r$, $\varphi$, and equations (\[eq:Calogero-1\]), (\[eq:Calogero-2\]) and (\[eq:conditions\]) are transformed into $$\begin{split}
  V_{\pm} & = \frac{1}{2\sqrt{\omega}} \biggl[\left(\mp \cos \varphi \partial_r  \pm \frac{1}{r} \sin \varphi
        \partial_{\varphi} + \omega r \cos \varphi \mp \cos \varphi \partial_r W \pm \frac{1}{r} \sin \varphi
        \partial_{\varphi} W\right) {b^{\dagger}}_x \\
  & \quad + \left(\mp \sin \varphi \partial_r  \mp \frac{1}{r} \cos \varphi \partial_{\varphi} + \omega r 
        \sin \varphi \mp \sin \varphi \partial_r W \mp \frac{1}{r} \cos \varphi \partial_{\varphi} W\right) {b^{\dagger}}_y 
        \biggr], \\
  W_{\pm} & = \frac{1}{2\sqrt{\omega}} \biggl[\left(\mp \cos \varphi \partial_r  \pm \frac{1}{r} \sin \varphi
        \partial_{\varphi} + \omega r \cos \varphi \pm \cos \varphi \partial_r W \mp \frac{1}{r} \sin \varphi
        \partial_{\varphi} W\right) b_x \\
  & \quad + \left(\mp \sin \varphi \partial_r  \mp \frac{1}{r} \cos \varphi \partial_{\varphi} + \omega r 
        \sin \varphi \pm \sin \varphi \partial_r W \pm \frac{1}{r} \cos \varphi \partial_{\varphi} W\right) b_y 
        \biggr],
\end{split}  \label{eq:polar}$$ $$\begin{split}
  K_0 & = K_{0,{\rm B}} + 