  A multidimensional gravitational model on the manifold $M = M_0 \times
\prod_{i=1}^{n} M_i$, where M_i are Einstein spaces ($i \geq 1$), is studied.
For $N_0 = dim M_0 > 2$ the $\sigma$ model representation is considered and it
is shown that the corresponding Euclidean Toda-like system does not satisfy the
Adler-van-Moerbeke criterion. For $M_0 = R^{N_0}$, $N_0 = 3, 4, 6$ (and the
total dimension $D = dim M = 11, 10, 11$, respectively) nonsingular spherically
symmetric solutions to vacuum Einstein equations are obtained and their
generalizations to arbitrary signatures are considered. It is proved that for a
non-Euclidean signature the Riemann tensor squared of the solutions diverges on
certain hypersurfaces in $R^{N_0}$.
