,$$ where $\ell= 2m^2 / M^3$ is the crossover length scale characterizing the theory. The ‘$\pm$’ signs give the two different branches of braneworld solutions which, to some extent, are connected with the two different ways in which the Schwarzschild–(anti)-de Sitter bulk space is bounded by the brane: the inner normal to the brane can point either in the direction of increasing or decreasing of the bulk radial coordinate. Following [@SS], we refer to the model with the lower sign (‘$-$’) as to BRANE1, and to that with the upper sign (‘$+$’) as to BRANE2.

In the formal limit of $m \to 0$, the BRANE1 model reduces to the well-known expression obtained in the Randall–Sundrum case $$\label{rs-limit}
H^2 + \frac{\kappa}{a^2} = \frac{(\rho + \sigma)^2}{9 M^6} + \frac{\Lambda_{\rm
b}}{6} + \frac{C}{a^4} \, ,$$ while the BRANE2 model does not exist in this limit. This can be viewed as a simple consequence of the fact that Eq. (\[1st\]), being quadratic with respect of the Hubble parameter $H$ in the case $m \ne 0$, reduces to a linear equation with respect to $H$ in the formal limit $m \to 0$.

Static solutions {#static}
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We proceed to the investigation of static solutions in 