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

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.