  We consider the geodesic motion on the symmetric moduli spaces that arise
after timelike and spacelike reductions of supergravity theories. The geodesics
correspond to timelike respectively spacelike $p$-brane solutions when they are
lifted over a $p$-dimensional flat space. In particular, we consider the
problem of constructing \emph{the minimal generating solution}: A geodesic with
the minimal number of free parameters such that all other geodesics are
generated through isometries. We give an intrinsic characterization of this
solution in a wide class of orbits for various supergravities in different
dimensions. We apply our method to three cases: (i) Einstein vacuum solutions,
(ii) extreme and non-extreme D=4 black holes in N=8 supergravity and their
relation to N=2 STU black holes and (iii) Euclidean wormholes in $D\geq 3$. In
case (iii) we present an easy and general criterium for the existence of
regular wormholes for a given scalar coset.
