Integrability and Linearizability of the Lotka–Volterra System with a Saddle Point with Rational Hyperbolicity Ratio

In this paper, we consider normalizability, integrability and linearizability properties of the Lotka–Volterra system in the neighborhood of a singular point with eigenvalues 1 and −λ. The results are obtained by generalizing and expanding two methods already known: the power expansion of the first integral or of the linearizing transformation and the transformation of the saddle into a node. With these methods we find conditions that are valid for λ + or λ . These conditions will allow us to find all the integrable and linearizable systems for λ= and with p +.