with M, N, = 0 , ... 4 and κ 2 = M − 3 P , where M$_{P}$ is the 5 d reduced Planck mass, and Λ denotes the five-dimensional cosmological constant. We now review the results of [12]. To begin, we shall constrain the magnitude of the scalar field to lie at | Φ | = η , leading to the effective action where θ is the phase of Φ. The equations of motion for this model are where is the energy momentum tensor. We will look for a solution of the form where µ, ν = 0 , 1 , 2 , 3 denote the 4 d coordinates, and the compact extra dimension is pa- rameterized by the coordinate 0 ≤ y < 2 π . We are interested in the case i.e., in solutions with the extra dimension stabilized at a constant circumference 2 πL . We shall also require that the four-dimensional slices are described by a spacetime of maximal symmetry whose scalar curvature is given by R (4) = 12 H $^{2}$, with H 2 negative for the anti-de Sitter (AdS) case. The solutions to the equations of motion for the scalar field Eq. (9) which are compatible with maximal spacetime symmetry are given by The change of axion phase θ around the compact dimension must be an integer multiple of 2 π , and hence the various flux vacua are parameterized by the integer n . S = ∫ d $^{5}$x √ − g$^{(}$ 1 2 κ $_{2}$R − 1 2 η $^{2}$∂$_{M}$ θ∂ $^{M}$θ − Λ$^{)}$ , (8) ∂$_{M}$ $_{(}$√ − g∂ $^{M}$θ ) = 0 , (9) R$_{AB}$ − 1 2 g$_{AB}$R = κ $^{2}$T$_{AB}$ , (10) T$_{AB}$ = η 2 ( ∂$_{A}$θ∂$_{B}$θ − 1 2 g$_{AB}$∂$_{M}$ θ∂ $^{M}$θ$^{)}$ − g$_{AB}$ Λ (11) ds 2 = g$_{MN}$ dx $^{M}$dxN = g$_{µν}$dx $^{µ}$dxν + g$_{yy}$ ( x $^{µ}$) dy $^{2}$, (12) g$_{yy}$ ( x $^{µ}$) = L 2 = const. , (13) θ ( x $^{M}$) = n y . (14) 4 A. The flux vacua