recent works. We propose here an efficient method for identifying such solutions without any [*a priori*]{} assumption on their shape. We find that the set of dynamically accessible self-similar solutions is in fact limited to one single object (as Nakano’s results suggested it). This proves that multiscaling properties should not be ascribed to the existence of a large manifold of singular behaviours. The exponent $z$ controlling the multiplicative growth of these particular solutions, accounts in a satisfying way for the asymptotic scaling properties of high order velocity or energy transfer moments. It is easy to extend this analysis to the GOY model, where basically the same conclusions concerning the unicity of solutions can be drawn. However self-similar solutions in that case are very mild and do not seem to play a major role in the statistics at high orders.\
The paper is organized as follows : in Section 2 we specify the conventions used in our computations for normalizing variables and parameters and describe some general properties of the ON model. Section 3 presents statistical results obtained from numerics. The emphasis is put on scaling exponents of the moments of energy transfer and their evolution with the relative proportion of 