Integrability and linearizability for Lotka-Volterra systems with the 3:−q resonant saddle point

Integrability and linearizability of a Lotka-Volterra system in a neighborhood of the singular point with eigenvalues 3 and any negative integer −q are studied completely. By computing the singular point quantities and generalized period constants, we obtain, respectively, the integrable and linearizable necessary conditions for this class of systems. Then we apply some effective ways to prove the sufficiency. Here the algorithms of finding necessary conditions are all linear and readily done using computer algebra system such as Mathematica or Maple, and these play an important role in solving completely the integrability and linearizability for the 3:−q resonant case.