An indirect method of finding integrals for three-dimensional quadratic homogeneous systems

Local integrability and linearizability of three-dimensional Lotka–Volterra systems

“…Other work on 3D Lotka-Volterra equations has been done by Bobienski and Żo ladek [3], who consider the finite singularity away from the axes planes, and give a number of mechanisms for the existence of a center in the (i : −i : λ) case; Cairo and Llibre [4,5], who obtain a number of conditions for the existence of Darboux first integrals in terms of the parameters; and Basov and Romanovski [1], who take one of eigenvalues equal to zero. There has also been several works devoted to systems which are homogeneous (λ = µ = ν = 0) and hence reducible to a two-dimensional Lotka Volterra equation ( [13,17,21,6]).…”

Integrability and linearizability of a family of three-dimensional quadratic systems

“…In the particular case where the axes planes x = 0, y = 0 and z = 0 are invariants of system (1) then we obtain a three dimensional Lotka-Volterra system. Such systems have been invistigated by several authors like Bobienski andŻo ladek [5], Aziz [1], Aziz and Christopher [3], Cairo and Llibre [10,11], Buzzi et al [9], Murza and Teruel [31], Basov and Romanovski [4], Christodoulides and Damianou [12], Gao and Liu, [19], Gonzalez and Peralta [20], Moulin-Ollagnier [30]) among others.…”

An integrating factor matrix method to find first integrals

“…Ollagnier [13] has found polynomial first integrals of the ABC system. Gao and Liu [14] presented a method that basically relies on changing variables to transform three-dimensional LV systems to two-dimensional ones. The existence of first integrals follows from integrating the two-dimensional systems.…”