–2 0 2 4 6 8 10 1 1.2 1.4 1.6 1.8 2 rho k = –1 –2 0 2 4 6 8 0.2 0.4 0.6 0.8 1 1.2 1.4 r k = 0 –2 0 2 4 6 8 0.2 0.4 0.6 0.8 1 1.2 1.4 r FIG. 2: f $^{2}$( r ) versus r for b = 1, c = 2 . 5, l = 0 . 5 and n = 5. α = 0 . 1 (solid line: naked singularity), α = α$_{ext}$ (bold line: extreme black brane), α = 0 . 8 (dashed line: null singularity with two horizons), and α = 1 . 2 (dotted line: null singularity with one horizon), where α$_{ext}$ = 0 . 59 , 0 . 67 for k = − 1 , 0, respectively. FIG. 1: f $^{2}$( ρ ) versus ρ for b = 1, c = 1 . 5, l = 0 . 5, n = 5 and α = 0 (timelike singularity). k = 0 (bold line), and k = − 1 (dashed line). as r → ∞ ( ρ → ∞ ). One should note that for nonzero α the singularity may be null, while it is timelike for α = 0 (see Figs. 1 and 2). Also, figure 2 shows that depending on the metric parameters, these real solutions may be interpreted as black brane solutions with inner and outer horizons, an extreme black brane or naked singularity. 7