5 different gauges, defined, respectively, by Eqs. (3.1) and (3.11). In fact, in the framework of the HL theory the two different gauges represent two different theories - metric (3.1) represents a HL theory with projectability condi- tion, while metric (3.11) represents a HL theory without projectability condition. The LMP solutions were originally found for the HL theory with detailed balance but without projectability conditions in the form (3.9). There are two classes of solutions, given, respectively, by B. The LMP Solutions FIG. 1: The functions α$_{±}$ ( λ ) defined by Eq. (3.16). all the three scalars K, K$_{ij}$K ij and R are finite. When α = 0, it is for the solution (3.15), which is singular only when α$_{±}$ ≤ 2 at r = 0. From Fig.1 we can see that α$_{−}$ is always less than two for a finite λ . Therefore, this branch of solutions is always singular at the center. α$_{+}$ is always less than two for 1 / 3 ≤ λ < 1 and greater than two for λ > 1. Therefore, R$_{orth}$ is finite for the α$_{+}$ solutions with λ > 1 for any r , including the center r = 0, while it is singular at the center for the α$_{+}$ solutions with 1 / 3 ≤ λ < 1. with α being an arbitrary real constant. Fig. 1 schemati- cally shows the curves of α$_{±}$ vs λ , which shows that there is no solution for the α$_{+}$ branch when λ = 1. Solutions given by Eqs. (3.14) and (3.15) shall be re- ferred to, respectively, as Class A and B solutions. Par- ticular values of α$_{±}$ are Transforming the above solutions into the canonical ADM form (3.1), we find that for the solution (3.14), we have 1. Class A Solutions where in writing the above expressions, we had chosen Ψ = 0. Then, the corresponding three scalars are given by which are useful in the following discussions. In the orthogonal gauge (3.9), as mentioned previously, the extrinsic curvature K$_{ij}$ vanishes identically, while the 3-dimensional curvature for the above two classes of so- lutions can be written as which are all finite. As shown in [38], this solution is also a vacuum solution of the SVW generalization (1.7) with a non-vanishing constant curvature k = 6Λ$_{W}$ . Since it does not depend explicitly on the coupling constant ξ , it is a vacuum solution for any given ξ , including ξ = 0. For detail, we refer readers to Sec. V of [38]. for any Ψ( r ), and where x ≡ √ − Λ$_{W}$ r , and Φ = − 1 2 ln ( 1 + x $^{2}$) , (3.14) Φ = − 1 2 ln ( 1 + x 2 − αx α ± ) , Ψ = − β$_{±}$ ln( x ) + 1 2 ln ( 1 + x 2 − αx α ± ) , (3.15) α$_{−}$ ≡ 2 λ − √$_{6}$$_{λ}$ − 2 λ − 1 = 2(2 λ − 1) 2 λ $_{+}$√$_{6}$$_{λ}$ − 2 , α$_{+}$ ≡ 2 λ $_{+}$√$_{6}$$_{λ}$ − 2 λ − 1 , β$_{±}$ ≡ 2 α$_{±}$ − 1 , (3.16) α$_{−}$ ( λ ) =     $^{−}$$^{1}$$^{,}$ λ = 1 / 3, 0 , λ = 1 / 2, 1 / 2 , λ = 1, 2 / 3 , λ ≃ 1 . 375, 1 , λ = 3, 2 , λ = ∞ , α$_{+}$ ( λ ) =    $^{−}$$^{1}$$^{,}$ λ = 1 / 3, −∞ , λ = 1 − $^{0}$, + ∞ , λ = 1 $^{+0}$, 2 , λ = ∞ , (3.17) µ = −∞ , ν = − 1 2 ln ( 1 − Λ$_{W}$ r 2 ) , (3.19) K = K$_{ij}$K ij = 0 , R = 6Λ$_{W}$ , (3.20) R$_{orth}$ = 2 r 2 { α ( 1 + α$_{±}$ ) x α ± − 3 x 2 } . (3.18) When α = 0, the corresponding R$_{orth}$ is for the solution (3.14), which reduces to a constant. That is, in this case