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

We investigate the local integrability and linearizability of three dimensional Lotka–Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1,-1,1),(2,-1,1) and (1,-2,1)(1,-2,1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.