In this subsection, we apply the method of [@CS] to the case of braneworld theory with the scalar-curvature term in the action for the brane, i.e., where $m \ne 0$. We introduce the notation $$\begin{array}{l}
\displaystyle B=\frac{\Lambda_{\rm b}}{3 \ell^2 H^4}\, , \quad P
=\frac{\rho}{6m^2H^2}\, ,
\quad S=\frac{\sigma}{6m^2H^2}\, , \bigskip \\
\displaystyle L=\frac{1}{\ell^2H^2}\, , \quad K=\frac{\kappa}{a^2H^2}\, , \quad
D=\frac{C}{\ell^2 a^4H^4}\, ,
\end{array}$$ and express the deceleration parameter $$q=-\frac{\ddot{a}}{H^2a}$$ from the second-order differential equation (\[second\]): $$\label{acc}
q = \frac{L(1+K) -B +[P(1+3w)-2S] [P + S-
\frac{1}2(1+K)]}{L+P+S-\frac{1}2(1+K)}\, .$$

Equation (\[1st\]) implies the constraint $$\label{constraint1}
\left[ 1 + K - 2 (P + S) \right]^2 = 4 L \left( 1 + K \right) - 2 B - D \, .$$

Note that the introduced variables are infinite for $H=0$ and $m=0$ and, therefore, cannot describe these particular cases. The second case was already considered in the previous subsection and in [@CS], and the first case will be considered below in Sec. \[static-case\] (using different variables).

Introducing the prime derivative $$'=\frac{1}{H}\frac{d}{dt}\, ,$$ we obtain the system of equations describing the evolution of our dimensionless parameters: $$\label{system}
\begin{array}{l}
P'=-P(1+3w-2q)\, , \smallskip \\
S'=2S(1+q)\, , \smallskip \\
L'=2L(1+q)\, , \smallskip \\
B'=4B(1+q)\, , \smallskip \\
K'=2Kq\, , \smallskip 