  The modified Dirac equations for the massive particles with the replacement
of the physical mass $m$ with the help of the relation $m\rightarrow m_1+
\gamma_5 m_2$ are investigated. It is shown that for a fermion theory with a
$\gamma_5$-mass term, the limiting of the mass specter by the value $ m_{max}=
{m_1}^2/2m_2$ takes place. In this case the different regions of the unbroken
$\cal PT$ symmetry may be expressed by means of the restriction of the physical
mass $m\leq m_{max}$. It should be noted that in the approach which was
developed by C.Bender et al. for the $\cal PT$-symmetric version of the massive
Thirring model with $\gamma_5$-mass term, the region of the unbroken $\cal
PT$-symmetry was found in the form $m_1\geq m_2$ \cite{ft12}. However on the
basis of the mass limitation $m\leq m_{max}$ we obtain that the domain $m_1\geq
m_2$ consists of two different parametric sectors: i) $0\leq m_2 \leq
m_1/\sqrt{2}$ -this values of mass parameters $m_1,m_2$ correspond to the
traditional particles for which in the limit $m_{max}\rightarrow \infty$ the
modified models are converting to the ordinary Dirac theory with the physical
mass $m$; ii)$m_1/\sqrt{2}\leq m_2 \leq m_1$ - this is the case of the unusual
particles for which equations of motion does not have a limit, when
$m_{max}\rightarrow \infty$. The presence of this possibility lets hope for
that in Nature indeed there are some "exotic fermion fields".
  As a matter of fact the formulated criterions may be used as a major test in
the process of the division of considered models into ordinary and exotic
fermion theories. It is tempting to think that the quanta of the exotic fermion
field have a relation to the structure of the "dark matter".
