the zero-fermion states (\[eq:zero-fermion\]), whereas the remaining odd generators $V_{\pm}$ may lead to one- and two-fermion states. After some straightforward calculations, equations (\[eq:V-W-bis\]) and (\[eq:w-f\]) yield $$\begin{split}
  & V_+ |\tau, \tau + N, q\rangle \\
  & \quad = \frac{{\cal N}_{N,n}}{\sqrt{z}} \Bigl\{\Bigl[ [N + (n+a+b)k + 1]
         Z^{(2n+a+b)}_N(z)  - (N+1) Z^{(2n+a+b)}_{N+1}(z)\Bigr] \\
  & \quad \times \Phi^{(a,b)}_n(\varphi) {\bar{b}^{\dagger}}_x - (n+a+b)k Z^{(2n+a+b)}_N(z) 
         \Phi^{(a+1,b+1)}_{n-1}(\varphi) {\bar{b}^{\dagger}}_y\Bigr\} |0\rangle, \\
  & V_- |\tau, \tau + N, q\rangle \\
  & \quad = \frac{{\cal N}_{N,n}}{\sqrt{z}} \Bigl\{\Bigl[ (N+nk) Z^{(2n+a+b)}_N(z)  
         - [N + (2n+a+b)k] Z^{(2n+a+b)}_{N-1}(z)\Bigr] \\
  & \quad \times \Phi^{(a,b)}_n(\varphi) {\bar{b}^{\dagger}}_x + (n+a+b)k Z^{(2n+a+b)}_N(z) 
         \Phi^{(a+1,b+1)}_{n-1}(\varphi) {\bar{b}^{\dagger}}_y\Bigr\} |0\rangle, \\
  & V_+ V_- |\tau, \tau + N, q\rangle = - V_- V_+ |\tau, \tau + N, q\rangle \\
  & \quad = {\cal N}_{N,n} (n+a+b)k Z^{(2n+a+b)}_N(z) \Phi^{(a+1,b+1)}_{n-1}(\varphi) {\bar{b}^{\dagger}}_x {\bar{b}^{\dagger}}_y
         |0\rangle.
\end{split}  \label{eq:V-appl}$$ From (\[eq:def-1\]), it is obvious that the three states in (\[eq:V-appl\]) are eigenstates of $K_0$ and $Y$ with eigenvalues $\left(\tau + N + \frac{1}{2}, q + \frac{1}{2}\right)$, $\left(\tau + N - \frac{1}{2}, q + \frac{1}{2}\right)$ and $(\tau + N, q + 1)$, respectively.

These one- and two-fermion states can be easily normalized by using the Hermiticity properties and anticommutation relations of the ${\rm osp}(2/2, {{\mathbb{R}}})$ generators. The results read $$\begin{split}
  & |+, 