The N-species integrable Volterra system as a maximally superintegrable Hamiltonian system

TL;DR

This paper demonstrates that the N-species Volterra system is not only integrable but also maximally superintegrable, reducible to a single degree of freedom, with supporting analytical and numerical evidence.

Contribution

It establishes the maximal superintegrability of the N-species Volterra system, extending previous work on its integrability and providing new analytical and numerical insights.

Findings

The N-species Volterra system is maximally superintegrable.

The system can be reduced to a single degree of freedom.

Analytical and numerical results support the superintegrability.

Abstract

The results presented in this paper are a natural development of those described in the paper {\it The Volterra Integrable case. Novel analytical and numerical results} (OCNMP Vol.4 (2024) pp 188-211), where the authors reconsidered the integrable case of the Hamiltonian $N$-species Lotka-Volterra system, introduced by Vito Volterra in 1937. There, an alternative approach for constructing the integrals of motion has been proposed, and compared with the old Volterra approach. Here we go beyond, and show that in fact the model introduced by Volterra and studied by us is not just integrable, but is maximally superintegrable and reducible to a system with only one degree of freedom regardless the number of species considered. We present both analytical and numerical results.