  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 free fermion theory with
a $\gamma_5$ mass term, the finiteness of the mass spectrum at the value $
m_{max}= {m_1}^2/2m_2$ takes place. In this case the region of the unbroken
$\cal PT$-symmetry may be expressed by means of the simple restriction of the
physical mass $m\leq m_{max}$. Furthermore, we have that the areas of unbroken
$\cal PT$-symmetry $m_1\geq m_2\geq 0$, which guarantees the reality values of
the physical mass $m$, consists of three different parametric subregions: i)
$0\leq m_2 < m_1/\sqrt{2}$, \,\,ii) $m_2=m_1/\sqrt{2}=m_{max},$
\,\,(iii)$m_1/\sqrt{2}< m_2 \leq m_1$. It is vary important, that only the
first subregion (i) defined mass values $m_1,m_2,$ which correspond to the
description of traditional particles in the modified models, because this area
contain the possibility transform the modified model to the ordinary Dirac
theory. The second condition (ii) is defined the "maximon" - the particle with
maximal mass $m=m_{max}$. In the case (iii) we have to do with the unusual or
"exotic" particles for description of which Hamiltonians and equations of
motion have no a Hermitian limit. 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".
