and decomposes into four ${\rm sp}(2, {{\mathbb{R}}}) \times {\rm so}(2)$ irreps $(\tau) (q)$, $(\tau - \frac{1}{2}) (q + \frac{1}{2})$, $(\tau + \frac{1}{2}) (q + \frac{1}{2})$ and $(\tau) (q + 1)$.

We conclude that the eigenstates of any supersymmetrized TTW Hamiltonian may be separated into basis states of an infinite collection of ${\rm osp}(2/2, {{\mathbb{R}}})$ irreps, each member of the set being characterized by a given value of the quantum number $n$ associated with the angular part of $H_k$ and determining the eigenvalue $(2n+a+b)^2 k^2$ of the first integral of motion $X_k$ [@tremblay09; @cq10a]. The corresponding eigenvalues of the second- and third-order Casimir operators [@frappat] $$\begin{split}
  C_2 &= K_0(K_0-1) - Y(Y+1) - K_+ K_- + V_- W_+ - V_+ W_-, \\
  C_3 &= (K_0 + Y)(K_0 - Y -1)(Y + \tfrac{1}{2}) - (Y + \tfrac{1}{2}) K_+ K_- + \tfrac{1}{2}[K_- V_+ - (K_0 
         - 3Y) V_-] W_+ \\
  & \quad + \tfrac{1}{2}[K_+ V_- - (K_0 + 3Y) V_+] W_-
\end{split}$$ are given by $n(n+a+b)k^2$ and $- \frac{1}{2} (a+b)n(n+a+b)k^3$, respectively. They vanish for $n=0$ in agreement with a known property of atypical LWS irreps. It is worth mentioning that similar relationships between Casimir operators and the angular part of Hamiltonians have already been 