Families of invariants of the motion for the Lotka–Volterra equations: The linear polynomials family

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Abstract
The modified Carleman embedding method already introduced by the authors to find first integrals (invariants of the motion) of polynomial form to the Lotka–Volterra system is described in detail, and its efficiency to treat the N-dimensional system proved. Using this method, an extensive investigation is performed for polynomials of the first degree, which allow a classification of the integrals in three families. For some systems possessing one invariant it is possible to find a second invariant using rescaling methods. They represent very restrictive solutions, implying that there exists a great number of conditions among the equation’s coefficients to satisfy. A proof is given that the Volterra invariants can be deduced as a limit. Finally, the interesting properties of the solutions of these systems are studied in detail.