sufficient conditions stating that isotropisation needs $d\beta_\pm/dt\rightarrow 0$ and considering the field equations, we recover two of the four conditions of the Collins and Hawking definition for isotropy. It shows the physical meaning of the limit $p_\pm e^{3\Omega}\rightarrow 0$ regarding isotropisation. Moreover, assuming that $\beta_\pm$ tend toward some constants means that the matrix $\beta$ whose components are the $\beta_{ij}$ becomes a constant $\beta_0$ which is the fourth condition defining isotropy in [@ColHaw73].\
In what follows, we study isotropic equilibrium states of each Bianchi class A model with curvature.

The Bianchi type $II$ model {#s11}
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### Field equations {#s111}

To study the equilibrium points corresponding to asymptotic isotropic states, we will use the following variables, specific to the Bianchi type $II$ model: $$\label{v1}
x_\pm=p_\pm H^{-1}$$ $$\label{v2}
y=\pi R_0^3 \sqrt{U}e^{-3\Omega}H^{-1}$$ $$\label{v3}
z=p_{\phi}\phi(3+2\omega)^{-1/2}H^{-1}$$ $$\label{v4}
w=\pi R_0^2 e^{-2\Omega+2(\beta_++\sqrt{3}\beta_-)}H^{-1}$$ They are independent since $x_\pm$, $y$, $z$ and $w$ are respectively functions of the independent quantities $p_\pm$, $\phi$, $p_\phi$ and $\beta_\pm$. Then, the Hamiltonian (\[hamiltonien\]) yields: $$\label{contrainte}
x_+^2+x_-^2+24y^2+12z^2+12w^2=1$$ We will consider this last expression as a constraint. It shows that the 5 variables $(x_\pm,y,z,w)$ are normalised. They allow us to rewrite the field equations as a first order equations system in the following way: $$\label{eq1}
