  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. 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 till the general $N-$species model.
