1 0 r (r) α < α c c r s 7 (r) r 0 α = α c c c r r 1 2 1 0 r (r) c α < α c r$_{s}$ FIG. 3: The function Δ( x ) defined by Eq. (3.22) for the α + branch with λ > 1, where α$_{c}$ is the value of α , for which Δ $^{′}$( x ) > 0 for α < α$_{c}$ . K and K$_{ij}$K ij diverge at r = r$_{s}$ , while R diverges as r → ∞ . FIG. 5: The function Δ( x ) defined by Eq. (3.22) for the α$_{−}$ branch with 1 < λ < 3, where α$_{c}$ is the value of α , for which Δ( x ) ≥ 0 is possible only when α > α$_{c}$ . K and K$_{ij}$K ij diverge at the two points r = r 1 , 2 , while R remains finite there. FIG. 4: The function Δ( x ) defined by Eq. (3.22) for the α$_{−}$ branch with 1 / 3 ≤ λ < 1, where α$_{c}$ is the value of α , for which Δ $^{′}$( x ) < 0 for α < α$_{c}$ . R, K and K$_{ij}$K ij diverge at the center r = 0, while only K and K$_{ij}$K ij diverges at r = r$_{s}$ . is non-negative. At the two points r = r$_{1}$ and r$_{2}$ we have Δ( r$_{i}$ ) = 0, and Eq. (3.23) shows that both K and K$_{ij}$K ij become unbounded at these points, while R re- mains finite. Therefore, in the present case there are also two scalar curvature singularities, located, respectively, at r = r$_{1}$ and r = r$_{2}$ for α > α$_{c}$ . Solutions with α ≤ α$_{c}$ are not physical. Case vi) β = β$_{−}$ = 1 , λ = 3 : In this case we have α$_{−}$ = β$_{−}$ = 1, and both r = 0 and r = r$_{s}$ . At the center, all three scalars R, K and K$_{ij}$K ij become unbounded, while only K and K$_{ij}$K ij diverge at r = r$_{s}$ . Case v) β = β$_{−}$, 1 < λ < 3 : In this case we have 1 / 2 ≤ α$_{−}$ < 1 , 0 ≤ β$_{−}$ < 1, and Δ = α x − 1 x 2 = { −∞ , x = 0, 0 , x → ∞ . (3.30) Clearly, to have Δ ≥ 0, we must assume α > 0. Then, for r ≥ r$_{s}$ ≡ 1 / $_{(}$√ − Λ$_{W}$ α ), Δ is non-negative [See Fig.6, Curve (a)]. At r = r$_{s}$ , Δ( r ) = 0, and Eq. (3.23) shows that both K and K$_{ij}$K ij become unbounded at this point, while R remains finite. As r → ∞ , Δ( r ) → 0, and K and K$_{ij}$K ij become unbounded again, while R still remains finite. Therefore, in the present case there are also two scalar curvature singularities, located, re- spectively, at r = r$_{s}$ and r = ∞ . Case vii) β = β$_{−}$, λ > 3 : In this case we have α$_{−}$ > 1 , β$_{−}$ > 1, and Δ = { −∞ , x = 0, +1 , x → ∞ . (3.31) Δ = { −∞ , x = 0, , x → ∞ . (3.29) Fig.6(b) shows the general properties of Δ( r ), from which we can see that for any given λ , there always exists a point r = r$_{s}$ where Δ( r$_{s}$ ) = 0. When r > r$_{s}$ , Δ is always positive. Eq. (3.23) shows that now both K and K$_{ij}$K ij diverge at r = r$_{s}$ , while all three scalars become unbounded as r → ∞ . Thus, in the present case there are two scalar curvature singularities, located, respectively, at r = r$_{s}$ and r = ∞ . Fig.5 shows the general properties of Δ( r ), from which we can see that for any given λ , there always exists a crit- ical value α$_{c}$ , for which Δ( r ) ≥ 0 is possible only when α > α$_{c}$ . In the latter case, Δ( r ) = 0 always has two positive roots, say, r$_{1}$ and r$_{2}$ . Without loss of general- ity, we assume r$_{2}$ > r$_{1}$ > 0. When r$_{1}$ ≤ r ≤ r$_{2}$ , Δ( r )