  We study some aspects of dynamical compactification scenario where
stabilisation of extra dimensions occurs due to presence the Gauss-Bonnet term
and non-zero spatial curvature. In the framework of the model under
consideration there exists two-stages scenario of evolution of a Universe: on
the first stage, the space evolves from a totally anisotropic state to the
state with 3-dimensional (corresponding to our "real", world) expanding and
$D$-dimensional contracting isotropic subspaces; on the second stage, constant
curvature of extra dimensions begins to play role and provide compactification
of extra dimensions. It is already known that such a scenario is realizable
when constant curvature of extra dimensions is negative. Here we show that a
range of coupling constants for which exponential solutions with 3-dimensional
expanding and $D$-dimensional contracting isotropic subspaces are stable is
located in a zone where compactification solutions with positively curved extra
space are unstable, so that two-stage scenario analogous to the one described
above is \emph{not} realizable. Also we study "nearly-Friedmann", regime for
the case of arbitrary constant curvature of extra dimensions and describe new
parametrization of the general solution for the model under consideration which
provide elegant way of describing areas of existence over parameters space.
