Trees and superintegrable Lotka-Volterra families

TL;DR

This paper introduces a family of superintegrable Lotka-Volterra systems associated with trees, demonstrating their integrability and reduction to solvable lower-dimensional systems.

Contribution

It establishes a novel connection between tree structures and superintegrable Lotka-Volterra systems, including their reduction and solvability.

Findings

Systems are superintegrable for generic parameters

Reduction to lower-dimensional superintegrable systems

Explicit integrals of motion identified

Abstract

To any tree on $n$ vertices we associate an $n$-dimensional Lotka-Volterra system with $3n-2$ parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits $n-1$ functionally independent integrals. We also show how these systems can be reduced to an ($n-1$)-dimensional system which is superintegrable and solvable by quadratures.