Developing a non-Hermitian algebraic theory with the $\gamma_5$-extension of mass

We consider a non-Hermitian fermionic model with the $\gamma_5$-extension of mass: $m\to m_1+\gamma_5m_2$. We establish the relation between this theory and a geometric theory with the maximum mass in the quantum mechanical approximation. We propose a more detailed condition of $PT$-symmetry preservation in the theory. It implies segregating the initial domain of $PT$-symmetry preservation into subdomains corresponding to descriptions of standard and exotic particles. We calculate the operator $\mathcal C$ in the new scalar product in such a theory with a non-Hermitian Hamiltonian and describe some consequences of introducing this operator. We find the eigenvalues and eigenvectors of this Hamiltonian.