that whatever the way we use such that the system be autonomous, equilibrium states are only determined by zeros of the system (\[eq1\]-\[eq5\]). The same reasoning may be applied for each Bianchi model. In what follows, we will use (\[eq1\]-\[eq5\]) to get the expressions for equilibrium points as some functions of $\ell(\phi)$ and then $\dot{\phi}(z,w)$ to get the asymptotical behaviour of the scalar field.

### Isotropic equilibrium states {#s112}

*Equilibrium points*\
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Several equilibrium points exist and we have to select those representing an isotropic equilibrium state. Immediately, we observe that the necessary conditions for isotropisation $d\beta_\pm/d\Omega\rightarrow 0$, imply that $x_\pm\rightarrow 0$ near equilibrium[^1]. Since the equations (\[betapm\]) and the definitions of the variables $x_\pm$ will be identical for all the Bianchi models, these limits will be the same for each of them. Thus, we will systematically discard any equilibrium point with $x_\pm\not =0$ or which is not defined by real values.\
All the equilibrium points of the Bianchi type $II$ model are summarised in the appendix. The first one is defined by $(y,w)=(0,0)$. Let us show that it is not consistent with isotropy. From equations (\[ppp\]-\[pmp\]), we derive that $p_+-\sqrt{3}p_-=p_0$, $p_0$ being an integration constant. Then, using (\[betapm\]), it 