should vary as $p_{0\pm}e^{-4\Omega}$ ($p_{0\pm}$ being some constants) for Bianchi type $VI_0$ or even slower for the $VII_0$ type since $\beta_\pm$ might tend toward some vanishing constants. It would follow that $p_\pm\rightarrow p_{1\pm}$, $p_{1\pm}$ being some integration constants. However, in this case $\dot{\beta}_\pm$ would tend asymptotically toward the constants $p_{1\pm}H^{-1}$ and isotropisation could not occur for a diverging value of $\Omega$. Hence, $(y,w_+,w_-)=(0,0,0)$ is not compatible with isotropisation except for the special case of zero measure $p_{1\pm}=0$.\
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*Monotonic functions*\
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What about monotonic functions? We can rewrite equation (\[hamp\]) as follows: $$\label{hamp2}
\dot{H}=-H\left[72y^2+24(w_+\pm w_-)^2\right]$$ As for the Bianchi type $II$ model, (\[hamp2\]) shows that the Hamiltonian is a monotonic function of constant sign which determines if isotropisation occurs at early or late times depending if the Hamiltonian is initially negative or positive. Moreover, it follows that $y$ and $w_\pm$ are of constant sign.\
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*Asymptotic behaviours*\
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Making the same approximation as for subsection \[s112\], we find that near an isotropic equilibrium state, $w_\pm\rightarrow e^{(1-\ell^2)\Omega}$. Then, assuming that isotropisation arises for $\Omega\rightarrow -\infty$ as we will show it below, it follows from (\[hamp2\]) that $H\rightarrow e^{-2\Omega}$ when $\ell\rightarrow 1$ and then, near equilibrium, $w_\pm$ should tend toward some non 