the first model. Lastly, the GOY model admits a second quadratic invariant besides kinetic energy, which is thought to play a prominent role in fixing its statistical properties [@KLWB; @GPZ]. Such an extra invariant is definitely absent in the ON model.\

Fixed points and qualitative description of the phase diagram
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First of all, let us say a few words about the existence and the nature of the fixed points of the model, which lead to Kolmogorov scaling. In the absence of dissipation and for an infinite number of shells, $\phi_{n}=1$ is an obvious solution for any value of $\beta$. However, once viscous dissipation is introduced, it is easy to see that static solutions making physical sense can exist only for $\beta \neq 0$. Indeed for $\alpha =1$ (OG chain), one has to solve for every $n>0$ $$Q^{2n\over 3}(\phi_{n-1}\phi_{n} - \phi_{n+1}^2)=\frac{Q^{2n}}{R} \phi_{n}
\label{pfobu_n}$$ In order to balance the leading order terms of this equation for $n \rightarrow
+\infty$, $\phi_{n}$ must behave like $-\displaystyle{Q^{4n\over 3}\over R}$ which is not acceptable. This absence of any fixed point in the presence of dissipation for the Obukhov chain is consistent with the dynamical behaviour one observes in this case. A solitary wave of negative 