  A classical model of gravity theory with several dilatonic scalar fields and
differential forms admitting an interpretation in terms of intersecting
p-branes is studied in (pseudo)-Riemannian space-time $M =R_+\times
S^{d_0}\times R_t\times M_2^{d_2}...\times M_n^{d_n}$ of dimension D. The
equations of motion of the model are reduced to the Euler-Lagrange equations
for the so-called pseudo-Euclidean Toda-like system. We suppose that the
characteristic vectors related to the configuration of p-branes and their
couplings to the dilatonic scalar fields may be interpreted as the root vectors
of a Lie algebra of the types $A_r, B_r, C_r$. In this case the model is
reducible to one of the open Toda chain's algebraic generalization and is
completely integrable by the known methods. The corresponding general solutions
are presented in explicit form. The particular exact solution describing a
class of nonextremal black holes is obtained and analyzed.
