The generation of magnetic field, together with the electrostatic activity met in the saturation regime of the Weibel instability (WI), is investigated by means of an analytical multi-stream model in a Hamiltonian framework. Taking advantage from the invariance of the generalized canonical momentum, the model allows to reduce the full kinetic 1D2V Vlasov equation into several 1D1V equations while keeping its kinetic character. The multi-stream model provides a more complete and accurate picture of the Weibel instability, because it is possible to separate the specific contribution of each stream during the development of the Weibel instability. An interesting result for the multi-stream mode is a lower cost in the perpendicular treatment of the p y momentum component since no differential operator associated with some approximate numerical scheme has to be carried out on this variable. Indeed, a small number of streams or particle classes are sufficient to correctly describe the magnetic field generation and the mixed electrostatic-electromagnetic nature of the instability.