to ease coordinates separability of (8), we obtain It is obvious that separability is granted through a variety of choices. The simplest of which may be sought in an obviously ”manifested-by-equation (12)” general identity of the form In this case, we may avoid any specifications on the forms of g ( ρ ), f ( ϕ ), and k ( z ) rather than being mathematically and quantum mechanically ”very well” defined. However, the energy term, 2 g ( ρ ) f ( ϕ ) k ( z ) E , in (12) suggests three feasible separabilities for f ( ϕ ) = 1 = k ( z ), k ( z ) = 1 = g ( ρ ), and f ( ϕ ) = 1 = g ( ρ ). We focus on one of these cases in the sequel. Let us consider the position-dependent-mass function to be only an explicit function of ρ . Namely, we choose f ( ϕ ) = 1 = k ( z ) and g ( ρ ) = ρ − 2 so that M ( ρ, ϕ, z ) = M ( ρ ) = ρ − $^{2}$. Under these settings, equation(12) collapses into a simple separable form 6 0 = [ Φ ′′ ( ϕ ) Φ ( ϕ ) + 2 E − ˜ V ( ϕ ) + 2 ( ζ − β − 1) ] + ρ 2 [ R ′′ ( ρ ) R ( ρ ) + 3 ρ R ′ ( ρ ) R ( ρ ) − ˜ V ( ρ ) + Z ′′ ( z ) Z ( z ) − ˜ V ( z ) ] . (14) 2 MV ( ρ, ϕ, z ) = 2 g ( ρ ) f ( ϕ ) k ( z ) V ( ρ, ϕ, z ) = ˜ V ( ρ ) + ˜ V ( z ) + 1 ρ 2 ˜ V ( ϕ ) . (13) 0 = 2 g ( ρ ) f ( ϕ ) k ( z ) [ E − V ( ρ, ϕ, z )] + [ R ′′ ( ρ ) R ( ρ ) − ( g ′ ( ρ ) g ( ρ ) − 1 ρ ) R ′ ( ρ ) R ( ρ ) + ζ 2 ( g ′ ( ρ ) g ( ρ ) ) 2 − ( β + 1) 2 ( g ′ ( ρ ) ρg ( ρ ) + g ′′ ( ρ ) g ( ρ ) )] + [ Z ′′ ( z ) Z ( z ) − k ′ ( z ) k ( z ) Z ′ ( z ) Z ( z ) + ζ 2 ( k ′ ( z ) k ( z ) ) 2 − ( β + 1) 2 k ′′ ( z ) k ( z ) ] + 1 ρ 2 [ Φ ′′ ( ϕ ) Φ ( ϕ ) − f ′ ( ϕ ) f ( ϕ ) Φ ′ ( ϕ ) Φ ( ϕ ) + ζ 2 ( f ′ ( ϕ ) f ( ϕ ) ) 2 − ( β + 1) 2 f ′′ ( ϕ ) f ( ϕ ) ] (12)