spinor. It is these two fundamental properties supplied by the deformed bosonic oscillator [@vas] that will help us to ‘extract the square root’ from the basic equations (\[kle\]) and (\[maj\]), and to construct a minimal covariant set of linear differential equations for fractional spin fields.

So, let us consider the algebra [@vas] $$\begin{aligned}
&[a^{-},a^{+}]=1+\nu K,&\label{hei}\\
&K^{2}=1,\quad Ka^{\pm}+a^{\pm}K=0,&
\label{klein}\end{aligned}$$ with the mutually conjugate operators $a^{-}$ and $a^{+}$, $(a^{-})^{\dagger} =a^+$, self-conjugate Klein operator $K$ and real deformation parameter $\nu$. In the undeformed case ($\nu=0$), eq. (\[klein\]) can be considered simply as a relation defining the Klein operator $K$.

Under application of algebra (\[hei\]), (\[klein\]) to the quantum mechanical 2-body Calogero model, the Klein operator $K$ is considered as an independent operator being an operator of the permutation of two identical particles, whereas the creation-annihilation operators $a^\pm$ are constructed from the relative coordinate and momentum operators of the particles, $$a^{\pm}=\frac{q\mp ip}{\sqrt{2}},
\label{apq}$$ and in representation where the operator of the relative coordinate $q$ to be diagonal, the momentum operator is realized in the form [@bri1]–[@bri2]: $$p=-i\left(\frac{d}{dq}-\frac{\nu}{2q}K\right).
\label{pdK}$$ Therefore, in such a case algebra (\[hei\]), (\[klein\]) is an extension of the Heisenberg algebra. On the other hand, one can realize operator $K$ in terms 