About the non-integrability in the Friedmann-Robertson-Walker cosmological model

We study the non integrability of the Friedmann-Robertson-Walker cosmological model, in continuation of the work [5] of Coehlo, Skea and Stuchi. Using Morales-Ramis theorem ([10]) and applying a practical non-integrability criterion deduced from it, we find that the system is not completely integrable for almost all values of the parameters lambda and lambda, which was already proved by the authors of [5] applying Kovacic's algorithm. Working on a level surface H = h with h ¹ 0 and h ¹ -1/4\lambda and using the Morales-Ramis-Simo ''higher variational'' theory ([11]), we prove that the hamiltonian system cannot be integrable for particular values of lambda among the exceptional values and that it is completely integrable in two special cases (lambda = lambda = -m² and lambda = lambda = m^2/3). We conjecture that there is no other case of complete integrability and give detailed arguments towards this