1 Introduction tonians possess properties similar to the PT -symmetric ones (here the non-Hermitian P$_{ϕ}$T$_{ϕ}$ -symmetric Hamiltonians) are nicknamed as pseudo- PT -symmetric. PACS codes: 03.65.Ge, 03.65.Ca Keywords: Non-Hermitian Hamiltonians, spherical-separablility, pseudo- PT -symmetry. In the search for the reality conditions on the energy spectra/eigenvalues of non-Hermitian Hamiltonians [1-35], it is nowadays advocated (with no doubts) that the orthodoxal mathematical Hermiticity requirement to ensure the re- ality of the spectrum of a Hamiltonian is not only fragile but also physically deemed remote, obscure and strongly unnecessary. A tentative weakening of the Hermiticity condition through Bender’s and Boettcher’s [1] PT -symmetric quantum mechanics (PTQM) (with P denoting parity and T time-reversal, a Hamiltonian H is PT -symmetric if it satisfies PT H PT = H ) has offered an alternative axiom that allows for the possibility of non-Hermitian Hamiltonians. Such a PTQM theory, nevertheless, has inspired intensive research on the non-Hermitian Hamiltonians and led to the so-called pseudo-Hermitian Hamil- tonians (i.e., a pseudo-Hermitian Hamiltonian H satisfies η H η − 1 = H † or η H = H $^{†}$η , where η is a Hermitian invertible linear operator and ( $^{†}$) denotes the adjoint) by Mostafazadeh [16-21] which form a broader class of non-Hermitian Hamiltonians with real spectra and encloses within those PT -symmetric ones. Moreover, not restricting η to be Hermitian (cf., e.g., Bagchi and Quesne [33]), and linear and/or invertible (cf., e.g., Solombrino [28], Fityo [29], and Mustafa and Mazharimousavi [30-32]) would weaken pseudo-Hermiticity and lead to real spectra. 2