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 nonintegrability criterion deduced from it, we find that the system is not completely integrable for almost all values of the parameters X and A, which was already proved by the authors of [5] applying Kovacic's algorithm. Working on a level surface H = h with h not equal 0 and h not equal -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 =-m(2) and lambda = Lambda=-m(2)/3). We conjecture that there is no other case of complete integrability and give detailed arguments towards this.