the Klein-Gordon equation $$(P^{2}+m^{2})\Psi=0
\label{kle}$$ and linear differential equation $$(PJ-sm)\Psi=0.
\label{maj}$$ Here $P_\mu =-i\partial_\mu$, the metric is $\eta_{\mu\nu}=diag(-1,1,1)$, and operators $J^{\mu}$, $\mu=0,1,2,$ being the generators of the group $\overline{\rm
SL(2,R)}$, form the $sl(2)$ algebra $$[J_{\mu},J_{\nu}]=-i\epsilon_{\mu\nu\lambda}J^{\lambda},
\label{alg}$$ where $\epsilon_{\mu\nu\lambda}$ is an antisymmetric tensor normalized so as $\epsilon^{012}=1$. We suppose that field $\Psi=\Psi^{n}(x)$ is transformed according to the one of the infinite-dimensional unitary irreducible representations (UIRs) of the group $\overline{\rm SL(2,R)}$: either of the discrete type series $D^{\pm}_{\alpha}$, or of the principal or supplementary continuous series $C^{\theta}_{\sigma}$ [@bar]. These representations are characterized by the value of the Casimir operator $J^2$ and by eigenvalues $j_0$ of the operator $J_0$. In the case of the representations $D^{\pm}_{\alpha}$, we have $J^2 =-\alpha(\alpha-1)$, $\alpha>0$, and $j_0 =\pm(\alpha+n)$, $n=0,1,2,\ldots,$ and, therefore, these representations are half-bounded. In the case of the representations of the continuous series $C^\theta_\sigma$, $J^{2}=\sigma$ and $j_0 =\theta+n$, $\theta\in [0,1)$, $n=0,\pm 1,\pm 2, \ldots,$ where $\sigma\geq
1/4$ for the principal series and $0<\sigma<1/4$, $\sigma>\theta(1-\theta)$ for the supplementary series [@bar].

The system of eqs. (\[kle\]) and (\[maj\]) has nontrivial solutions under the coordinated choice of the representation and parameter $s$, and as a result, these equations fix the values of the $\overline{\rm ISO(2,1)}$ Casimir operators, which 