Based on the inspiring example, nevertheless, by Bender, Brody and Jones [35] that H = p $^{2}$+ x $^{2}$+2 x is a non- PT -symmetric whereas a simple amendment H = p $^{2}$+( x + 1) $^{2}$− 1 (that leaves the Hamiltonian invariant and allows parity to perform reflection about x = − 1 rather than x = 0) would consequently classify H as PT -symmetric (i.e., reflection need not necessarily be through the origin) and promoting Znojil’s understanding [34] of Bender’s and Boettcher’s PTQM (i.e., P and T need not necessarily mean just the parity and time reversal, respectively), we may introduce [36] a time-reversal-like, and a parity-like operators that might very well be accommodated by Bender’s and Boettcher’s PTQM. In this case, P ϕ acting on a function f ( r, θ, ϕ ) ∈ L$_{2}$ would read P $_{ϕ}$f ( r, θ, ϕ ) = f ( r, θ, 2 π − ϕ ). Moreover, f ( r, θ, ϕ ) is said to be P ϕ T $_{ϕ}$-symmetric if it satis- fies P ϕ T $_{ϕ}$f ( r, θ, ϕ ) = f ( r, θ, ϕ ). Hence, our new operators leave the coordi- nates r and θ unaffected and are designed to operate only on the azimuthal descendent eigenvalue equation (e.g., equation (21) below) of the spherically- separable non-Hermitian Hamiltonians (4) and (5). However, it should be noted that our parity-like operator P ϕ in (2) is Hermitian, unitary, and per- forms reflection through a 2D-mirror represented by the xz -plane. Yet, the proof of the reality of the eigenvalues of a P ϕ T $_{ϕ}$-symmetric Hamiltonian is straightforward. Let the eigenvalue equation of our P ϕ T $_{ϕ}$-symmetric Hamil- tonian be Hψ ( r, θ, ϕ ) = Eψ ( r, θ, ϕ ), then P ϕ T $_{ϕ}$Hψ = P ϕ T $_{ϕ}$Eψ = Eψ . Using [ P ϕ T $_{ϕ}$, H ] = 0 we obtain Eψ = E $^{∗}$ψ and E is therefore pure real (in analogy 3 T ϕ : ̂ L$_{z}$ = − i∂/∂ϕ −→ − ̂ L$_{z}$ = i∂/∂ϕ, ϕ −→ ϕ, i −→ − i (1) P ϕ : ̂ L$_{z}$ −→ − ̂ L$_{z}$, ϕ −→ (2 π − ϕ ) , (2)