Integrable Multidimensional Cosmology for Intersecting $p$-Branes

A multidimensional field model describing the behaviour of (at most) one Einstein space of non-zero curvature and n Ricci-flat internal spaces is considered. The action contains several dilatonic scalar fields and antisymmetric forms. The problem setting covers various problems with field dependence on a single space-time coordinate, in particular, isotropic and anisotropic homogeneous cosmologies. When the forms are chosen to be proportional to volume forms of p-brane factor spaces, a Toda-like Lagrange representation arises. Exact solutions are obtained when the p-brane dimensions and the dilatonic couplings obey some orthogonality conditions. General features and some special cases of cosmological solutions are discussed. It is shown, in particular, that all hyperbolic models with a 3-dimensional external space possess an asymptotic with the external scale factor a(t) is proportional to |t| (the cosmic time), while all internal scale factors and all scalar fields tend to finite limits. For D=11 a family of models with one 5-brane and three 2-branes is described.