w_+)^2)&\\
&\dot{z}=y^2(72z-12\ell)+24z(w_-\pm w_+)^2&\\
&\dot{w}_+=2w_+\left[x_++\sqrt{3}x_-+12(w_-\pm w_+)^2+36y^2-1\right]\label{eq52}&\\
&\dot{w}_-=2w_-\left[x_+-\sqrt{3}x_-+12(w_-\pm w_+)^2+36y^2-1\right]\label{eq62}&\\\nonumber\end{aligned}$$ The $\pm$ symbols in equations (\[contrainte1\]-\[eq62\]) correspond respectively to the Bianchi type $VI_0$ and $VII_0$ models. For the first model, the constraint shows that the variables are normalised since $w_+$ and $w_-$ are positive. For the second one, because of the minus sign, both $w_+$ and $w_-$ could diverge. Then the constraint will be respected only if the sum $w_+-w_-$ tends toward a constant. We will show below that isotropy is only compatible with finite values of $w_\pm$. Consequently, the isotropic states we are looking for are reached for some bounded values of the variables whatever the Bianchi type $VI_0$ or $VII_0$ models. Now, we can also show that isotropisation of Bianchi type $VI_0$ and $VII_0$ models may not arise for a finite value of $\Omega$. Indeed, if $\Omega$ tends toward a constant when the proper time diverges, $d\Omega/dt$ vanishes and thus, from (\[lapse\]), it comes that $H$ vanishes. But then, $w_\pm$ should diverge which is not compatible with the equilibrium as shown above. Hence, for the Bianchi type $VI_0$ and $VII_0$ models, isotropisation does not lead to a static Universe.

### Isotropic equilibrium states {#s122}

*Equilibrium points*\
\
All the equilibrium points 