Integrability of Lotka-Volterra type systems of degree 4

We study the integrability of the Lotka-Volterra type systems with 1:−(3q−1) resonances. We prove some sufficient conditions for the integrabilities of the systems and give some necessary conditions by studying the first two saddle values of the system. In the particular cases of 1:−2, 1:−5 and 1:−8 resonances, we derive necessary and sufficient conditions for the integrabilities of the systems.