| description abstract | In this study, we investigate the integrability and linearizability problems of a family of
cubic three-dimensional Lotka–Volterra systems with one zero eigenvalue, involving
seventeen parameters. Necessary conditions on the parameters of the system for both
integrability and linearizability are obtained by computing the resonant quantities
using Gröbner bases and decomposing the variety of the ideal generated in the ring
of polynomials of parameters of the system. The sufficiency of these conditions is
also proven except that for a case, Case 32, of sufficiency has been left as conjectural.
In particular, we used the Darboux method, the existence of a first integral with an
inverse Jacobi multiplier, time reversibility, the properties of linearizable nodes in two
dimensional systems and power series arguments to the third variable and some other
techniques | en_US |
