  Multidimensional gravity interacting with intersecting electric and magnetic
$p$-branes is considered for fields depending on a single variable. Some
general features of the system behaviour are revealed without solving the field
equations. Thus, essential asymptotic properties of isotropic cosmologies are
indicated for different signs of spatial curvature; a no-hair-type theorem and
a single-time theorem for black holes are proved (the latter makes sense in
models with multiple time coordinates). The validity of the general
observations is verified for a class of exact solutions known for the cases
when certain vectors, built from the input parameters of the model, are either
orthogonal in minisuperspace, or form mutually orthogonal subsystems. From the
non-existence of Lorentzian wormholes, a universal restriction is obtained,
applicable to orthogonal or block-orthogonal subsystems of any $p$-brane
system.
