fluctuations carrying the system away from the K41 fixed point cannot grow from shell to shell more rapidly than $Q^{\gamma n}$. On the other hand, the fact that $\gamma$ takes a finite value even close to the transition suggest that such fluctuations are efficient as soon as the instability threshold is passed. It is the purpose of this Section to identify the set of singular fluctuations that the ON model can admit.\
Since we are now interested in nonlinear instabilities occuring in the inertial range, we may forget about forcing and dissipation, and think of the shell number $n$ as running from $-\infty$ to $+\infty$. Let us rewrite the equation of motion in terms of new variables $b_{n}=Q^{n}u_{n}=Q^{2n\over 3}\phi_{n}$ ($b_{n}$ is nothing dimensionally but the gradient of the velocity field). We get from (\[nlphi\_n\]) and (\[equphi\_n\]), after absorbing the factor $\alpha Q^{2\over 3}$ into a rescaling of time : $$\frac{d}{dt} b_{n}= N_{n}[\vec{b}]=(b_{n-1}b_{n}+{\beta\over \alpha}
Q^{2\over 3} b_{n-1}^2) - {1\over Q^{2}}(b_{n+1}^2+{\beta\over \alpha}
Q^{2\over 3}b_{n}b_{n+1})
\label{nlb_n}$$ Since ${\vec N}[\vec b]$ does not depend explicitely on $n$ and is quadratic in $\vec b$, the set of equations (\[nlb\_n\]), for $-\infty
< n< +\infty$, support formally self-similar solutions of the type : $$b_{n}(t)={1\over {t^{*}-t}} f(Q^{nz}(t^{*}-t))\;\equiv
