From this last expression and approximating (\[eq1\]) by $(3-\ell^2)x_+-24w^2$, we deduce that $x_+$ behaves as the sum of two terms $e^{2(1-\ell^2)\Omega}$ and $e^{(3-\ell^2)\Omega}$. Since isotropy needs $x_+\rightarrow 0$ and $\ell^2<3$, we derive it only occurs if $\Omega\rightarrow -\infty$ and $\ell^2< 1$. These two limits are in accordance with the vanishing of $x_\pm$ and $w$ which is necessary to reach the equilibrium isotropic state. Consequently $x_\pm$ asymptotically behave as $e^{2(1-\ell^2)\Omega}$. Let us note that the two limits $x_\pm\rightarrow 0$ and $\Omega\rightarrow -\infty$, necessary for isotropisation, are compatible and justify our assumption that it takes place for a diverging value of $t$: when $\Omega\rightarrow -\infty$ and the Universe isotropises, it is expanding and there is no physical meaning to consider that it ends for a finite value of the proper time.\
Let us show that the value $\ell^2=1$ has to be discarded. If $1-\ell^2\rightarrow 0$ faster than $\Omega^{-1}$, $w$ tends toward a non vanishing constant and is not compatible with isotropy. If $1-\ell^2\rightarrow 0$ slower than $\Omega^{-1}$, from (\[hamp\]) we deduce that near equilibrium $H\rightarrow e^{-2\Omega}$. Then from (\[ppp\]), it comes that $p_+\rightarrow e^{-2\Omega}$ and it follows from (\[v1\]) that $x_+$ tends toward a non vanishing constant. Hence, the limit $\ell^2\rightarrow 1$ 