characterized by the parameters $$\alpha_{+}=\frac{1}{4}(1+\nu)>0,\quad
\alpha_{-}=\alpha_{+}+\frac{1}{2}>\frac{1}{2}.
\label{aeo}$$ Relations (\[inv\]) and (\[irreo\]) mean that we have realized the UIRs of the discrete series $D^{+}_{\alpha_{+}}$ and $D^{+}_{\alpha_{-}}$ of the group $\overline{\rm SL(2,R)}$ on the even and odd subspaces of the total Fock space, spanned by the states $|k>_{+}$ and $|k>_{-}$, respectively. The UIRs of the discrete series $D^{-}_{\alpha_{\pm}}$ can be obtained from realization (\[jmu\]) by means of the obvious substitution $$J_{0}\rightarrow -J_{0},\quad
J_{\pm}\rightarrow -J_{\mp}.$$ Further on, for the sake of simplicity we shall consider only the case of representations $D^{+}_{\alpha}$.

It is necessary to note here, that the described realization of UIRs $D^{\pm}_{\alpha_{+}}$ and $D^{\pm}_{\alpha_{-}}$ generalizes the well known realization of the representations $D^{\pm}_{1/4}$ and $D^{\pm}_{3/4}$ in terms of the ordinary bosonic oscillator ($\nu=0$) [@per].

For the construction of the minimal spinor set of linear differential equations sought for, we introduce the $\gamma$-matrices in the Majorana representation, $$(\gamma^{0})_{\alpha}{}^{\beta}=-(\sigma^{2})_{\alpha}{}^{\beta},\quad
(\gamma^{1})_{\alpha}{}^{\beta}=i(\sigma^{1})_{\alpha}{}^{\beta},\quad
(\gamma^{2})_{\alpha}{}^{\beta}=i(\sigma^{3})_{\alpha}{}^{\beta},$$ which satisfy the relation $\gamma^{\mu}\gamma^{\nu}=-\eta^{\mu\nu}+i\epsilon^{\mu\nu\lambda}
\gamma_{\lambda}.$ Here $\sigma^{i}$, $i=1,2,3,$ are the Pauli matrices, and raising and lowering the spinor indices is realized by the antisymmetric tensor $\epsilon_{\alpha\beta}$, $\epsilon_{12}=\epsilon^{12}=1$: $f_{\alpha}=f^{\beta}\epsilon_{\beta\alpha},$ $f^{\alpha}=\epsilon^{\alpha\beta}f_{\beta}.$

Consider the spinor type operator $$\begin{aligned}
L_{\alpha}=
\left(\begin{array}{ccc}
q\\
p
\end{array}\right),
\label{lab}\end{aligned}$$ constructed from the operators $q$ and $p$, which, in turn, 