  Exact static, spherically symmetric solutions to the Einstein-Maxwell-scalar
equations, with a dilatonic-type scalar-vector coupling, in $D$-dimensional
gravity with a chain of $n$ Ricci-flat internal spaces are considered, with the
Maxwell field potential having two nonzero components: the temporal,
Coulomb-like one and the one pointing to one of the extra dimensions. The
properties and special cases of the solutions are discussed, in particular,
those when there are horizons in the space-time. Two types of horizons are
distinguished: the conventional black-hole (BH) ones and those at which the
physical section of the space-time changes its signature (the latter are called
{\it T-horizons}). Two theorems are proved, one fixing the BH- and T-horizon
existence conditions, the other claiming that the system under study cannot
have a regular center. The stability of a selected family of solutions under
spherically symmetric perturbations is studied. It is shown that only
black-hole solutions are stable, while all others, in particular, those with
T-horizons are unstable.
