but also by $V_-$. Hence it is an ${\rm osp}(2/2, {{\mathbb{R}}})$ LWS and the states obtained from it by means of the raising generators form a basis for a so-called atypical LWS irrep with $\tau = - q$ [@balantekin]. The latter is known to decompose into two ${\rm sp}(2, {{\mathbb{R}}}) \times {\rm so}(2)$ irreps characterized by $(\tau) (q)$ and $(\tau + \frac{1}{2}) (q + \frac{1}{2})$, respectively. This is confirmed by setting $n=0$ in the generic results (\[eq:V-appl\]) and (\[eq:normalized-fermionic\]). We indeed get $$|-, \tau + N + \tfrac{1}{2}, q + \tfrac{1}{2}\rangle = |+, \tau + N + \tfrac{1}{2}, q + \tfrac{1}{2}\rangle, \qquad
  |\pm, \tau + N, q + 1\rangle = 0$$ for $N=0$, 1, 2, …. In the $n=0$ case, we may therefore set $$|\tau + \tfrac{1}{2}, \tau + N + \tfrac{1}{2}, q + \tfrac{1}{2}\rangle = |+, \tau + N + \tfrac{1}{2}, q + 
  \tfrac{1}{2}\rangle, \qquad N=0, 1, 2, \ldots.$$ It is worth observing that since $Q$ and $Q^{\dagger}$ in (\[eq:SUSY\]) annihilate the ground state of ${\cal H}^s$, supersymmetry is unbroken.

For non-vanishing values of $n$, the situation is more complicated since the three states in (\[eq:V-appl\]) and (\[eq:normalized-fermionic\]) are non-zero. It is however straightforward to show that the one-fermion 