# Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families

## TL;DR

This paper introduces a method to construct superintegrable Lotka-Volterra systems with many parameters, applying it to systems with 2 to 5 components, and linking these systems to tree structures.

## Contribution

The paper develops a novel construction method for superintegrable Lotka-Volterra systems and establishes a correspondence with tree graphs for systems with up to five components.

## Key findings

- Constructed superintegrable families with 3n-2 parameters
- Established a one-to-one correspondence with trees on n vertices
- Presented explicit examples for systems with 2 to 5 components

## Abstract

We present a method to construct superintegrable $n$-component Lotka-Volterra systems with $3n-2$ parameters. We apply the method to Lotka-Volterra systems with $n$ components for $1 < n < 6$, and present several $n$-dimensional superintegrable families. The Lotka-Volterra systems are in one-to-one correspondence with trees on $n$ vertices.