b_i,
  \label{eq:Calogero-1}$$ where $W$ denotes some function of the $x_i$’s. Then the even operators $$\begin{split}
  & K_0 = K_{0,{\rm B}} + \Gamma, \qquad K_{\pm} = K_{\pm,{\rm B}} - \Gamma, \qquad Y = \frac{1}{4}
       \sum_i (2x_i \partial_i W + [{b^{\dagger}}_i, b_i]), \\
  & K_{0,{\rm B}} = D + \frac{1}{4} \omega \sum_i x_i^2, \qquad K_{\pm,{\rm B}} = - D
       + \frac{1}{4} \omega \sum_i x_i^2 \mp \frac{1}{4} \sum_i (2x_i \partial_i + 1), \\
  & D = \frac{1}{4\omega} \sum_i [- \partial_i^2 - \partial_i^2 W + (\partial_i W)^2], \qquad \Gamma =
       \frac{1}{2\omega} \sum_{ij} \partial_{ij}^2 W\, {b^{\dagger}}_i b_j,
\end{split}  \label{eq:Calogero-2}$$ resulting from the anticommutation relations in (\[eq:def-1\]), also satisfy the remaining defining relations of ${\rm osp}(2/2, {{\mathbb{R}}})$ provided $$\biggl[\sum_i x_i \partial_i, D\biggr] = - 2D, \qquad \biggl[\sum_i x_i \partial_i, \Gamma\biggr] = - 2\Gamma, 
  \qquad \sum_i x_i \partial_i W = C,  \label{eq:conditions}$$ where $C$ is some constant. This is the kind of realization that leads to the dynamical superalgebra of Calogero-type Hamiltonians [@freedman; @brink93; @brink98; @ghosh01] if $x_i$, $i=1$, 2, …, $\nu$, denote the coordinates of $\nu$ particles on a line and $W$ is an appropriate solution of equation (\[eq:conditions\]).

Realization of ${\rm osp}(2/2, {{\mathbb{R}}})$ in polar coordinates for the TTW Hamiltonians
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Let us consider the 