1 − m ˜ a 4 0 + q 2 ˜ a 6 0 ≥ 0 . (37) A 2 + B 3 ≥ 0 . (38) ( − π ˜ a 3 0 4 ˜ σ 0 ) 2 + ( − ˜ a 2 0 8 + 1 − f (˜ a 0 ) 2 ) 3 ≥ 0 (39) f (˜ a 0 ) − 2 + ˜ a 2 0 2 ≤ 0 . (40) 0 ≤ 1 − m ˜ a 4 0 + q 2 ˜ a 6 0 ≤ ( 4 ˜ a 2 0 − 1 ) 2 . (41) ˜ σ 0 > 0 . (42) [ 3 ˜ a 0 + 4 ˜ a 3 0 ( f (˜ a 0 ) − 3) ] < 0 (43) 0 ≤ 1 − m ˜ a 4 0 + q 2 ˜ a 6 0 < 4 ( 4 ˜ a 2 0 − 1 ) 2 . (44) 0 ≤ 1 − m ˜ a 4 0 + q 2 ˜ a 6 0 ≤ ( 4 ˜ a 2 0 − 1 ) 2 . (45) 8 ii ) In the potential, the reality condition requires also that At the location of the throat this amounts to or after some manipulation it yields This is equivalent to iii ) Our last constraint condition concerns, the positivity of the energy density, which means that This implies, from (31) that or equivalently It is remarkable to observe now that the foregoing constraints ( i − iii ) on our parameters can all be expressed as a single constraint condition, namely We plot ˜ V $^{′′}$(˜ a ) from (26) for various fixed values of mass and charge, as a projection into the plane with coordinates β and ˜ a 0 . In other words, we search and identify the regions for which ˜ V $^{′′}$(˜ a ) > 0, in 3 − dimensional figures considered as a projection in the ( β, ˜ a 0 ) plane. The metric function f ( r ) and energy density ˜ σ 0 > 0 , behavior also are given in Fig.s 1-4. It is evident from Fig.s 1-4 that for increasing charge the stability regions shrink to smaller