  We consider a D-dimensional cosmological model describing an evolution of
Ricci-flat factor spaces, M_1,...M_n (n > 2), in the presence of an m-component
perfect fluid source (n > m > 1). We find characteristic vectors, related to
the matter constants in the barotropic equations of state for fluid components
of all factor spaces.
  We show that, in the case where we can interpret these vectors as the root
vectors of a Lie algebra of Cartan type A_m=sl(m+1,C), the model reduces to the
classical open m-body Toda chain.
  Using an elegant technique by Anderson (J. Math. Phys. 37 (1996) 1349) for
solving this system, we integrate the Einstein equations for the model and
present the metric in a Kasner-like form.
