of supersymmetry breaking is defined by the deformation parameter $\nu$.

Due to the property $E_{n}^{-}>0$ taking place for $\epsilon=-$, we can construct the Fermi oscillator operators: $$\begin{aligned}
f^{\pm}&=&\frac{Q^{\mp}_{-}}{\sqrt{H_{-}}}
\label{fdef}\\
&=&a^{\pm}\cdot \frac{\Pi_{\pm}}{\sqrt{N+\Pi_{+}}},
\nonumber\end{aligned}$$ $$\{f^{+},f^{-}\}=1,\quad
f^{\pm 2}=0,$$ i.e. one can realize a Bose-Fermi transformation in terms of one bosonic oscillator. Let us note that though operators $a^{\pm}$ do not commute with $f^{\pm}$, $$[a^{\pm},f^{\pm}]\neq 0,$$ nevertheless, the operator $H_{\epsilon}$ can be written in the form of the simplest supersymmetric Hamiltonian of the superoscillator [@nic]: $$H_{\epsilon}=\frac{1}{2}\{a^{+},a^{-}\}+\epsilon \frac{1}{2}[f^{+},f^{-}].$$

Having fermionic oscillator variables (\[fdef\]), we can construct the hermitian operators: $$S_{1}=\frac{1}{2}(f^{+}+f^{-}),\quad
S_{2}=-\frac{i}{2}(f^{+}-f^{-}),\quad
S_{3}=f^{+}f^{-}-1/2.$$ They act in an irreducible way on every 2-dimensional subspace of states $(|2k>,|2k+1>)$, $k=0,1,\ldots$, and form $su(2)$ algebra: $$[S_{i},S_{j}]=i\epsilon_{ijk}S_{k}.$$ Due to the relation $S_{i}S_{i}=3/4,$ this means that we have realized spin-$1/2$ representation of the $su(2)$ algebra in terms of one bosonic oscillator in contrast to the well known Schwinger realization of Lie algebra $su(2)$ in terms of two bosonic oscillators [@schw]. Therefore, one can realize unitary spin-1/2 representation of SU(2) group on the Fock space of one bosonic oscillator.

Before passing over to the generalization of the constructions to the case of more complicated quantum mechanical supersymmetric systems, corresponding to the systems with 