  We consider a multidimensional model of the universe given as a
$D$-dimensional geometry, represented by a Riemannian manifold $(M,g)$ with
arbitrary signature of $g$, $M= \R\times M_1\times \cdots \times M_n$, where
the $M_i$ of dimension $d_i$ are Einstein spaces, compact for $i>1$. For
Lagrangian models $L(R,\phi)$ on $M$ which depend only on the Ricci curvature
$R$ and a scalar field $\phi$, there exists a conformal equivalence with
minimal coupling models. For certain nonminimal models we study classical
solutions and their relation to solutions in the equivalent minimal coupling
model. The domains of equivalence are separated by certain critical values of
the scalar field $\phi$. Furthermore, the coupling constant $\xi$ of the
coupling between $\phi$ and $R$ is critical at both, the minimal value $\xi=0$
and the conformal value $\xi_c=\frac{D-2}{4(D-1)}$. In different noncritical
regions of $\xi$ the solutions behave qualitatively different. Instability can
occure only in certain ranges of $\xi$. {This paper is dedicated to Prof. D. D.
Ivanenko.}
