Integrability of Lotka–Volterra type systems of degree 4

We study the integrability of the Lotka–Volterra type systems with 1 : − ( 3 q − 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.