be realised even in the spatially flat universe. It is the negative dark radiation (the constant $C$ in Eq. (\[first\])) which plays the crucial role in the dynamics of this model and which, therefore, also needs to be constrained by cosmological observations.

### Stability of the spatially flat static solutions

For $\kappa=0$, our equations (\[second0\]), (\[1st0\]) and (\[w\]) simplify to $${} -3M^{6}\Lambda_{\rm b} + \left[ \rho (1+3w) - 2 \sigma \right] \cdot
\left(\rho + \sigma \right)=0\, ,$$ $$(\rho + \sigma)^2 = {} - 9M^{6} \left( \frac{\Lambda_{\rm b}}{6} +
\frac{C}{a^4} \right)$$ and $$W=\frac{1}{1+\frac{2m^2}{3M^{6}}(\rho_0+\sigma)}\left[-(1+3w)
\frac{4C}{a_0^4}+\frac{3\sigma(1+w)}{\rho_0+\sigma} \frac{2C}{a_0^4}\right]\, ,$$ respectively.

Introducing the dimensionless parameters $$x = \frac{\rho_0\ell^2}{6m^2 }= \frac{2m^2\rho_0}{3M^{6}} \, , \quad y =
\frac{\sigma \ell^2}{6m^2} = \frac{2m^2\sigma}{3M^{6}} \, , \quad z =
\frac{\Lambda_{\rm b}\ell^2}{6} = \frac{2m^4\Lambda_{\rm b}}{3M^{6}}\, ,$$ we can express the quantities that we used before in their terms. Since only two of these parameters are independent, it is convenient to express the relevant quantities in terms of $x$ and $y$. We have $$z=\frac{1}2(x+y)\left[(1 + 3w)x - 2y \right]\, ,\label{z}$$ $$\label{C}
\frac{C}{a_0^4}=-\frac{3M^{6}}{8m^4} (1 + w) x(x + y)\, ,$$ $$\label{W}
W=\frac{3M^{6}}{4m^4}(1+w)x\frac{2(1+3w)x - (1-3w)y}{1+x+y}\, .$$

Taking into account the physical restrictions (\[rest1\]) and (\[rest2\]) and equations (\[z\]) and (\[W\]), we can indicate the 