assume that the relation between $\rho$ and $a$ is given by Eq. (\[1st0\]) at the initial moment of time.

We express the energy density $\rho$ through $a$ using Eq. (\[1st0\]), $$\rho=-\sigma+3m^2\frac{\kappa}{a^2}\mp
3M^3\sqrt{\frac{\kappa}{a^2}-\frac{\Lambda_{\rm b}}{6}-\frac{C}{a^4}}\, ,$$ and substitute it to (\[1st1\]): $$\label{1st2}
\frac{\ddot{a}}{a}=\frac{A_1}{9M^{6}\mp
18m^2M^3\sqrt{\frac{\kappa}{a^2}-\frac{\Lambda_{\rm b}}{6}-\frac{C}{a^4}}}\, ,$$ where $$A_1 = -9M^6 \left( \frac{\kappa}{a^2} - \frac{\Lambda_{\rm b}}3 \right) \pm
3M^3 A_2 \sqrt{\frac{\kappa}{a^2} - \frac{\Lambda_{\rm b}}{6} -
\frac{C}{a^4}}\, ,$$ $$A_2=-3\sigma (1+w)+3(1+3w)m^2\frac{\kappa}{a^2}\mp
3(1+3w)M^3\sqrt{\frac{\kappa}{a^2}-\frac{\Lambda_{\rm b}}{6}-\frac{C}{a^4}}\, .$$

After that, we introduce the perturbation of the static solution $$a=a_0(1-\epsilon)\, , \quad \epsilon (t) \ll 1 \, ,$$ where $a_0$ is a stationary solution of (\[1st2\]), and expand Eq. (\[1st2\]) around the stationary point: $$\ddot \epsilon = W \epsilon \, ,$$ where $$\label{w}
W = \frac{B_1}{1 \pm \frac{2m^2}{M^3} \sqrt{\frac{\kappa}{a_0^2} -
\frac{\Lambda_{\rm b}}{6} - \frac{C}{a_0^4}}}\, ,$$ $$\begin{array}{l}
\displaystyle B_1 = \frac{2\kappa}{a_0^2}+(1+3w)
\left(\frac{2\kappa}{a_0^2}-\frac{4C}{a_0^4}\right)\mp B_2\pm B_3 \pm B_4\, ,
\smallskip \\
\displaystyle B_2=\frac{\sigma(1+w)}{M^3
\sqrt{\frac{\kappa}{a_0^2}-\frac{\Lambda_{\rm
b}}{6}-\frac{C}{a_0^4}}}\left(\frac{\kappa}{a_0^2} -\frac{2C}{a_0^4}\right)\, ,
\smallskip \\
\displaystyle B_3=\frac{m^2}{M^3}\frac{2\kappa}
{a_0^2}(1+3w)\sqrt{\frac{\kappa}{a_0^2}-\frac{\Lambda_{\rm
b}}{6}-\frac{C}{a_0^4}}\, ,
\smallskip \\
\displaystyle B_4=\frac{m^2}{M^3}\frac{\kappa}{a_0^2}(1+3w)\frac{\frac{\kappa}
{a_0^2}-\frac{2C}{a_0^4}}{\sqrt{\frac{\kappa}{a_0^2}- \frac{\Lambda_{\rm
b}}{6}-\frac{C}{a_0^4}}}\, .
\end{array}$$ The lower and upper signs in these equations refer to the BRANE1 and BRANE2 models, respectively.

It can also be noted [@SS] that, unlike the general relativity theory, the braneworld theory allows for empty ($\rho = p = 0$) static 