On a Quadratic Poisson Algebra and Integrable Lotka – Volterra Systems with Solutions in Terms of Lambert’s $W$ Function

We study a class of integrable inhomogeneous Lotka – Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We provide the Poisson algebra of their Darboux polynomials and prove a contraction theorem. We then use these results to classify the systems according to the number of functionally independent (and, for some, commuting) integrals. We also establish separability/solvability by quadratures, given the solutions to the $2$- and $3$-dimensional systems, which we provide in terms of the Lambert $W$ function.