in terms of hidden nonlinear supersymmetries [@plyu96; @plyu00; @correa]. Furthermore, some ${\cal N}=4$ supersymmetric extensions have been recently considered in connection with the ${\rm su}(1,1/2)$ superalgebra [@wyllard; @gala07; @krivonos; @gala09] or more generally with $D(2,1;\alpha)$ [@hakobyan].

The purpose of the present paper is to show that the family of TTW Hamiltonians corresponding to any positive real value of $k$ admits a ${\cal N}=2$ supersymmetric extension including the standard one of Calogero-type Hamiltonians [@freedman; @brink93; @brink98; @ghosh01] as special case for $k=2$ and 3.

In section 2, we review some known realizations of the ${\rm osp}(2/2, {{\mathbb{R}}})$ superalgebra in cartesian coordinates. In section 3, we make a transformation to polar coordinates on a plane to deal with the case of the TTW Hamiltonians. Bases for the corresponding irreducible representations (irreps) are constructed in section 4. The $k=1$, 2 and 3 examples are treated in detail in section 5. Finally, section 6 contains the conclusion.

Realizations of the ${\rm osp}(2/2, {{\mathbb{R}}})$ superalgebra
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The ${\rm osp}(2/2, {{\mathbb{R}}})$ superalgebra is generated by eight operators, four even ones closing the ${\rm sp}(2, {{\mathbb{R}}}) \times {\rm so}(2)$ Lie algebra and four odd ones, which separate into two ${\rm sp}(2, {{\mathbb{R}}})$ spinors [@nahm; @scheunert77a; @scheunert77b; @balantekin; 