$BC_2$ model ($k=2$) [@olsha81; @olsha83] and the three-particle Calogero model [@calogero69] with some extra three-body interaction ($k=3$), initially introduced by Wolfes and also considered by Calogero and Marchioro (CMW model) [@wolfes; @calogero74]. The fact that the $k=2$ and $k=3$ cases belong to the family of Calogero-type Hamiltonians associated with root systems of classical Lie algebras [@olsha81; @olsha83] has surely contributed to the interest aroused by the TTW Hamiltonians.

Some years ago, a ${\cal N}=2$ supersymmetric extension of the $\nu$-particle Calogero model, based on a dynamical ${\rm osp}(2/2, {{\mathbb{R}}}) \sim {\rm su}(1,1/1)$ superalgebra [@nahm; @scheunert77a; @scheunert77b; @balantekin; @frappat], has been introduced by Freedman and Mende [@freedman]. Several aspects of this extension and of its generalization to other root systems than that of the $A_{\nu-1}$ Lie algebra have been reviewed in the literature (see, e.g., [@brink93; @brink98; @ghosh01]). These supermodels are exactly solvable and play an important role in many areas of physics, such as the study of superstrings, black holes, superconformal quantum mechanics, spin chains, etc.

Since then, there have been great theoretical advances in that field too. For instance, the nonuniqueness of the construction proposed by Freedman and Mende has been discussed [@ghosh04; @gala06]. Calogero-like models have also been analyzed 