The generalized deformed oscillator algebra (\[gener\]), containing as other particular cases the algebras of the $q$-deformed Arik-Coon [@qac] and Macfarlane-Biedenharn [@qmb] oscillators, was considered by Dascaloyannis [@qd], and, so, all these deformed bosonic oscillators have a general structure (see also ref. [@qmmp]).

Let us consider now the following set of three operators $J_{\mu}$, bilinear in the creation-annihilation operators of the $\nu$-deformed oscillator: $$J_{0}=\frac{1}{4}\{a^{+},a^{-}\},\quad
J_{\pm}=J_{1}\pm iJ_{2}=\frac{1}{2}(a^{\pm})^{2}.
\label{jmu}$$ These operators satisfy the algebra (\[alg\]) of the generators of $\overline{\rm SL(2,R)}$ group for any value of the deformation parameter $\nu$ given by eq. (\[restr\]), whereas the value of the Casimir operator $J_{\mu}J^{\mu}$ is given here by the relation $$J^2=-\hat{\alpha}(\hat{\alpha}-1),$$ where $$\hat{\alpha}=\frac{1}{4}(1+\nu K).
\label{hatal}$$ So, realization (\[jmu\]) gives a reducible representation of the group $\overline{\rm SL(2,R)}$ on the Fock space of the $\nu$-deformed bosonic oscillator. As we have pointed out, the total Fock space can be separated into two subspaces, spanned by even, $|k>_{+}$, and odd, $|k>_{-},$ states being distinguished by the Klein operator: $$K|k>_\pm =\pm |k>_\pm,$$ $$|k>_{+}\equiv |2k>,\quad
|k>_{-}\equiv |2k+1>,\quad
k=0,1,\ldots.$$ These subspaces are invariant with respect to the action of operators (\[jmu\]): $$\begin{aligned}
J_{0}|k>_\pm&=&(\alpha_\pm+k)|k>_\pm,\quad k=0,1,\ldots,
\nonumber\\
J_{-}|0>_\pm&=&0,\quad
J_{\pm}|k>_\pm\propto|k\pm 1>_\pm,\, k=1,\ldots,
\label{inv}\end{aligned}$$ and Casimir operator takes on them the constant values, $$J_{\mu}J^{\mu}|k>_\pm=-\alpha_\pm(\alpha_\pm-1)|k>_\pm,
\label{irreo}$$ 