  In this work, we are investigating the problem of integrability of Bianchi
class A cosmological models. This class of systems is reduced to the form of
Hamiltonian systems with exponential potential forms.
  The dynamics of Bianchi class A models is investigated through the
Euler-Lagrange equations and geodesic equations in the Jacobi metric. On this
basis, we have come to some general conclusions concerning the evolution of the
volume function of 3-space of constant time. The formal and general form of
this function has been found. It can serve as a controller during numerical
calculations of the dynamics of cosmological models.
  The integrability of cosmological models is also discussed from the points of
view of different integrability criterions. We show that dimension of phase
space of Bianchi class A Hamiltonian systems can be reduced by two. We prove
vector field of the reduced system is polynomial and it does not admit any
analytic, or even formal first integral.
