El títol de la versió pre-print de l'article és: Analytic integrability of the Bianchi Class B cosmological model

Bianchi models are cosmological models that describe space-times which are foliated by homogeneous hypersurfaces of constant time and are divided into two classes, Class A and Class B. There are many studies about the integrability of Class A. Here we study the integrability of Class B. For the homogeneous cosmological models of Class B, Einstein's system of differential equations reduces to a dynamical system of dimension seven according to Bogoyavlensky's approach. We show that in order to study the integrability of such systems it is sufficient to deal with homogeneous polynomial differential systems of dimension six. Concretely, Bianchi V is the simplest model and can be written as a homogeneous polynomial differential system of degree 2. Bianchi IV is dealt as a homogeneous polynomial differential system of degree 3 and the rest of the models, Bianchis III, V I and V II are of degree 5. Due to the fact that all Bianchi class B models have been reduced to homogeneous polynomial differential systems, the study of their analytic integrability reduces to analyze their homogeneous polynomial first integrals. We show that Bianchi model V admits polynomial first integral, and we prove that the corresponding homogeneous polynomial differential systems that represent models Bianchi IV , III, V I and V II do not admit polynomial first integrals.