\dot{x}_+=72y^2x_++24w^2x_+-24w^2$$ $$\label{eq2}
\dot{x}_-=72y^2x_-+24w^2x_- -24\sqrt{3}w^2$$ $$\label{eq3}
\dot{y}=y(6\ell z+72y^2-3+24w^2)$$ $$\label{eq4}
\dot{z}=y^2(72z-12\ell)+24w^2z$$ $$\label{eq5}
\dot{w}=2w(x_++\sqrt{3}x_-+12w^2+36y^2-1)$$ with $\ell=\phi U_\phi U^{-1}(3+2\omega)^{-1/2}$. Since we want to keep $\omega$ and $U$ undetermined, we will not explicit the form of $\ell(\phi)$. Then, the above system could not seem autonomous because $\ell=\ell(\phi)$ and thus it would be meaningless to look for its equilibrium point. However, it exists two possibilities such that it becomes autonomous. The first one is to consider $\ell$ as a function of $(x,y,z,w)$ rather than $\phi$. Such considerations are often used when one look for exact solutions of field equations: for instance, instead of considering the potential as a function of the scalar field, it can be easier to find exact solutions by assuming it depends on the metric functions[@Arb99]. However, in the general case, $\ell$ may not be written in this way in the whole range of $\phi$. The second possibility, which applies whatever $\ell$, is to consider an additional first order equation for $\phi$ which is derived from (\[phip\]): $\dot{\phi}(z,\phi)$. Then, the system is autonomous. However, the constraint equation shows that scalar field equilibrium is not necessary for isotropisation contrary to other variables: $\dot\phi$ and $\phi$ can diverge whereas isotropy asymptotically occurs. Hence, we conclude 