can be written on the brane at y = 0 as, and on the brane y = L , The minimal conditions of potential V$_{y}$$_{=0}$ are and ∂V$_{y}$ =0 ∂χ 2 , 3 = ∂V$_{y}$ =0 ∂ψ 1 , 3 = 0 are automatically satisfied. While the minimal conditions on the brane y = L are We left four independent equations for the four unknown v , v$_{χ}$ , v$_{ψ}$ , and v$_{ϑ}$ in the limit of the trilinear couplings going to be zero. Thus the configurations needed in our scenario can be realized with ad hoc constraints. [1] G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. M. Rotunno, Phys. Rev. Lett. 101 , 141801 (2008) [arXiv:0806.2649 [hep-ph]]; arXiv:0809.2936 [hep-ph]. [2] M. C. Gonzalez-Garcia and M. Maltoni, Phys. Rept. 460 , 1 (2008) [arXiv:0704.1800 [hep-ph]]; T. Schwetz, AIP Conf. Proc. 981 , 8 (2008) [arXiv:0710.5027 [hep-ph]]. 30 ∂V$_{y}$$_{=}$$_{L}$ ∂ϕ$_{i}$ = 2 v { µ 2 Φ + 2(3 λ Φ 1 + 4 λ Φ $_{3}$) v $^{2}$} = 0 ( i = 1 , 2 , 3) . (A19) ∂V$_{y}$$_{=0}$ ∂χ$_{1}$ = 2 v$_{χ}$ { µ 2 χ + ( λ ψχ 1 + ωλ ψχ 2 + ω $^{2}$λ ψχ ∗ 2 ) v 2 ψ + 2( λ χ 1 + λ χ $_{2}$) v 2 χ } = 0 , (A16) ∂V$_{y}$$_{=0}$ ∂ψ$_{2}$ = 2 v$_{ψ}$ { µ 2 ψ + 2 λ ψ $_{1}$v 2 ψ + λ ψχ 1 v 2 χ + ( ωλ ψχ 2 + ω $^{2}$λ ψχ ∗ 2 ) v 2 χ + λ $^{ψϑ}$v 2 ϑ } = 0 , (A17) ∂V$_{y}$$_{=0}$ ∂ϑ = 2 v$_{ϑ}$ { µ 2 ϑ + 2 λ $^{ϑ}$v 2 ϑ + λ $^{ψϑ}$v 2 ψ } = 0 , (A18) V (Φ , η ) = λ $^{η}$Q η 1 + λ Φ η 1 Q Φ η 1 + λ Φ η 2 Q Φ η 2 + λ Φ η 3 Q Φ η 3 + λ Φ η ∗ 3 Q Φ η 4 . (A15) V ( η ) = µ 2 $_{η}$B$_{5}$ + λ $^{η}$Q η 1 (A14) V (Φ) = µ 2 $_{Φ}$B$_{1}$ + λ Φ 1 Q Φ 1 + λ Φ 2 Q Φ 2 + λ Φ 3 Q Φ 3 + λ Φ 4 Q Φ 4 + iλ Φ 5 Q Φ 5 (A13) V ( χ ) = µ 2 $_{χ}$B$_{2}$ + λ χ $_{1}$Q χ 1 + λ χ $_{2}$Q χ 2 + λ χ $_{3}$Q χ 3 (A8) V ( ψ ) = µ 2 $_{ψ}$B$_{3}$ + λ ψ $_{1}$Q ψ 1 + λ ψ $_{2}$Q ψ 2 + λ ψ $_{3}$Q ψ 3 (A9) V ( ϑ ) = µ 2 $_{ϑ}$B$_{4}$ + λ $^{ϑ}$Qϑ 1 (A10) V ( ψ, χ ) = λ ψχ 1 Q ψχ 1 + λ ψχ 2 Q ψχ 2 + λ ψχ ∗ 2 Q ψχ 3 + λ ψχ 3 Q ψχ 4 (A11) V ( ψ, ϑ ) = λ $^{ψϑ}$Q ψϑ 1 (A12)