In the present paper we reconsider the integrable case of the Hamiltonian $N$-species Volterra system, as it has been introduced by Vito Volterra in 1937 and significantly enrich the results already published in the ArXiv in 2019 by two of the present authors (M. Scalia and O. Ragnisco). In fact, we present a new approach to the construction of conserved quantities and comment about the solutions of the equations of motion; we display mostly new analytical and numerical results, starting from the classical predator-prey model and arriving at the general $N$-species model