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In this work, we are investigating the problem of integrability of Bianchiclass A cosmological models. This class of systems is reduced to the form ofHamiltonian systems with exponential potential forms.  The dynamics of Bianchi class A models is investigated through theEuler-Lagrange equations and geodesic equations in the Jacobi metric. On thisbasis, we have come to some general conclusions concerning the evolution of thevolume function of 3-space of constant time. The formal and general form ofthis function has been found. It can serve as a controller during numericalcalculations of the dynamics of cosmological models.  The integrability of cosmological models is also discussed from the points ofview of different integrability criterions. We show that dimension of phasespace of Bianchi class A Hamiltonian systems can be reduced by two. We provevector field of the reduced system is polynomial and it does not admit anyanalytic, or even formal first integral.Comment: 19 pages, Latex 2.0

On the integrability of Bianchi cosmological models