return to the discussion of these equations and connection (\[spvec\]) in last section.

As it was pointed out in the original papers [@vas], the deformed Heisenberg algebra (\[hei\]) and (\[klein\]) leads to different higher spin superalgebras. We have shown in the previous section that the $\nu$-deformed bosonic oscillator permits to generalize the well known representation of the $osp(1|2)$ superalgebra, realized by the ordinary bosonic oscillator ($\nu=0$) and characterized by the value of the Casimir operator $C=1/16$, to the representations with $C=\frac{1}{16}(1-\nu^2)$. Further, we have seen that the Klein operator introduces $Z_2$-grading structure on the Fock space of the ordinary or $\nu$-deformed bosonic oscillator. Such a structure is an essential ingredient of supersymmetry. Therefore, we arrive at the natural question: whether it is possible to use this $Z_2$-grading structure for realizing representations of $N=2$ supersymmetry in terms of $\nu$-deformed or ordinary ($\nu=0$) bosonic oscillator, and, therefore, for bosonizing supersymmetric quantum mechanics.

In the present section we shall investigate the problem of bosonization of supersymmetric quantum mechanics [@nic; @wit] in the systematic way. We shall consider the $\nu$-deformed bosonic oscillator, but it is necessary to stress that every time one can put $\nu=0$, and therefore, all the constructions will also be valid 