domains of existence of static solutions (both stable and unstable) in the $(x,y)$ plane.

The natural condition $\rho_0 > 0$ leads to the restriction $x > 0$.

For BRANE1, restrictions (\[rest1\]) additionally give $$y > 0\, , \quad 4y > (x + y) \left[ (1 + 3w) x - 2 y \right] \, , \quad (1 +
3w) x > (1 - 3w) y \, .$$

For BRANE2, restrictions (\[rest2\]) additionally give $$y > -1 \, , \quad 4y + 2 > (x + y) \left[ (1 + 3w) x - 2y \right] \, .$$

The corresponding domains are shown below in Figs. \[f:1\] and \[f:2\] for the two important cases $w = 0$ and $w = 1/3$. The shaded regions in these figures correspond to unstable static solutions ($W > 0$), and the dotted regions correspond to stable static solutions ($W \le 0$). In general, stable static solutions exist only for the BRANE2 model in the case $w < 1/3$.

It also follows from (\[C\]) that spatially flat static solutions with matter satisfying the strong energy condition can exist in the BRANE2 model with $C = 0$, i.e., with zero dark-radiation term, in contrast with the Randall–Sundrum model, where 