Q^{nz} g(Q^{nz}(t^{*}-t))
\label{selfsim_1}$$ In the equation above, $t^{*}$ is the critical time at which, in the absence of dissipation, the fluctuation reaches the end of the cascade. The scaling exponent $z$ is [*a priori*]{} arbitrary. However $z={2\over 3}$ gives back Kolmogorov scaling, while $z=1$ corresponds to the extreme situation of a fluctuation carrying a constant energy. One expects therefore ${2\over 3}\leq z \leq 1$ on physical grounds.\
Self-similar solutions, if they exist, are obviously good candidates for describing the growth of singular fluctuations. The question then arises whether many values of $z$ are dynamically accessible, which would be a natural source of multifractality, or whether on the contrary a single $z$ is selected. In that case, one should check that $\gamma =(z-{2\over 3})$, since moments $\Sigma_{n,p}$ are dominated by extreme fluctuations for high values of $p$. Self-similar solutions together with their exponent $z$ have already been determined by Nakano for the ON model [@N88]. He used a rather cumbersome iterative method to find them and we were not convinced he had exhausted the whole set of possibilities in his work. This is why we came back to this problem and were led to develop a procedure to be described below, 