vanishing constants. Thus the value $\ell=1$ does not agree with isotropisation.\
Concerning $x_\pm$, from (\[eq12\]-\[eq22\]), we deduce that they asymptotically behave as the sum of two exponentials $e^{2(1-\ell^2)\Omega}$ and $e^{(3-\ell^2)\Omega}$, showing again that these quantities will tend toward zero only if $\Omega\rightarrow -\infty$ and $\ell^2<1$. These two limits allow vanishing of $w_\pm$, which is necessary to reach the equilibrium states.\
All these elements show that near isotropy $w_\pm$ is bounded. Indeed, for $(x,y,z)$ to reach equilibrium, one only needs that $w_+-w_-\rightarrow 0$ whatever the particular asymptotical behaviours of $w_\pm$. Then only using this last limit, we have recovered the asymptotical behaviours for $x_\pm$ which implies that $\ell^2<1$. From (\[eq52\]-\[eq62\]) and these last expressions and condition, it is then possible to get the particular behaviours of $w_\pm$, which show that, near isotropy, these variables always vanish and are then bounded.\
Again, we are able to calculate that $p_\pm e^{3\Omega}$ tends to $0$ as $e^{(2-\ell^2)\Omega}$, thus filling the necessary but not sufficient conditions for isotropisation we defined above. From equation (\[hamp2\]) we derive that asymptotically $H$ behaves as $e^{(\ell^2-3)\Omega}$. The form of equation (\[phip\]) for $\dot{\phi}$ being unchanged whatever the Bianchi model as well as those of $\ell$ and $z$ near equilibrium, 