6 1 0 r (r) r$_{s}$ α < 0 2. Class B Solutions For Class B solutions (3.15), they can be written in the canonical ADM form (3.1) with where Clearly, to have real solutions, we must assume Δ ≥ 0. It should be noted that unlike Class A solutions, this class of solutions do not satisfy the vacuum equations of the HL theory with projectability condition, due to the fact that the HL actions are not invariant under the co- ordinate transformations (3.11). As shown in Appendix, they can be interpreted as solutions of the HL theory with projectability condition coupled with a spherical anisotropic fluid with heat flow. The properties of sin- gularities of these quantities can be well represented by the three scalars K, K$_{ij}$K ij and R , as can be seen from Appendix, so in the following we shall not consider them specifically. FIG. 2: The function Δ( x ) defined by Eq. (3.22) for the α + branch with 1 / 3 ≤ λ < 1. All three scalars R, K and K$_{ij}$K $^{ij}$, defined by Eq. (3.23), become unbounded at r = 0, while only K and K$_{ij}$K ij diverge at r = r$_{s}$ . µ = 1 2 ln Δ , ν = ( 1 − 2 α$_{±}$ ) ln( x ) , (3.21) Δ ≡ 1 − x 2(1 − 2 α ± ) − x 4(1 − α ± ) + αx 2 − 3 α ± . (3.22) Δ = { −∞ , x = 0, +1 , x → ∞ . (3.26) R = 2 r 2 { 1 − ( 1 + 2 β$_{±}$ ) x 2 β ± } , K = √ − Λ$_{W}$ 2Δ 1 / $^{2}$x3 − 2 α ± ( 4 √ − Λ$_{W}$ Δ − x $^{2}$δ ) , K$_{ij}$K ij = − Λ$_{W}$ 4Δ x 4(1 − α ± ) 8Δ − x $^{2}$δ$^{2}$) , (3.23) δ ≡ 2 ( 1 − 2 α$_{±}$ ) x 1 − 4 α ± + 4 ( 1 − α$_{±}$ ) x 3 − 4 α ± α ( 2 − 3 α$_{±}$ ) x 1 − 3 α ± . (3.24) Δ = { +1 , x = 0, −∞ , x → ∞ . (3.27) Δ = { 1 , x = 0, −∞ , x → ∞ . (3.25) Δ = αx 1 / 2 − x $^{2}$. (3.28) Case ii) β = β$_{+}$, λ > 1 : In this case we have α$_{+}$ ≥ 2 and β$_{+}$ ≥ 3, where the equality holds only when λ = ∞ . Then, from the expression (3.22) we find that Inserting the above solutions into Eq. (3.2) we find that where Fig.3 shows this case, from which we find that for any given λ > 1, there always exists a minimum r$_{s}$ > 0 so that Δ( r ) ≥ 0 for r ≥ r$_{s}$ , where r$_{s}$ is the solution of Δ( r ) = 0. Then, from the expressions (3.23) we find that K and K$_{ij}$K ij diverge at r = r$_{s}$ , while R diverges as r → ∞ . Therefore, in the present case there are also two scalar curvature singularities, located, respectively, at r = r$_{s}$ and r = ∞ . Case iii) β = β$_{−}$, 1 / 3 ≤ λ < 1 : In this case we have − 1 ≤ α$_{−}$ < 1 / 2 and − 3 β$_{−}$ < 0, where the equality holds only for λ = 1 / 3. Then, from the expression (3.22) we find that To study these solutions further, it is found convenient to consider the two branches α$_{±}$ separately. We shall use β = β$_{±}$ to denote the α$_{±}$ branches. Case i) β = β$_{+}$, 1 / 3 ≤ λ < 1 : In this case we have α$_{+}$ ≤ − 1 and β$_{+}$ ≤ − 3. Then, from the expression (3.22) we find that from Fig.4, we find that for any given λ in this range, there always exists a maximum r$_{s}$ > 0 so that Δ( r ) ≥ 0 for 0 ≤ r ≤ r$_{s}$ . Then, From the expressions (3.23) we find that all three scalars R, K and K$_{ij}$K ij become unbounded at the center, while only K and K$_{ij}$K ij di- verge at r = r$_{s}$ . Therefore, in the present case there are two scalar curvature singularities, located, respectively, at r = 0 and r = r$_{s}$ . Thus, there must exist a point x = x$_{s}$ at which we have Δ( x$_{s}$ ) = 0. Since Δ ≥ 0, we must restrict the solutions to the range 0 ≤ x ≤ x$_{s}$ (or equivalently, 0 ≤ r ≤ r$_{s}$ , where r$_{s}$ ≡ x$_{s}$$_{/}$√ − Λ$_{W}$ .). Fig. 2 shows this case. From the expressions (3.23) we find that all these three scalars diverge at the center r = 0, while K and K$_{ij}$K ij di- verge at r = r$_{s}$ . Therefore, in the present case there are two scalar curvature singularities, located, respectively, at r = 0 and r = r$_{s}$ . Case iv) β = β$_{−}$ = 0 , λ = 1 : In this case we have α$_{−}$ = 1 / 2, and Thus, to have Δ non-negative, we must assume α > 0. Then, Δ ≥ 0 for 0 ≤ r ≤ r$_{s}$ ≡ α 2 / $^{3}$, where Δ( x ) = 0 at