  We solve explicitly the geodesic equation for a wide class of
(pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G
compact, as well as non-compact semisimple Lie groups, under a simple algebraic
condition for the metric m. We prove that these manifolds are geodesically
complete and their geodesics are orbits in G/H of a product of N one-parameter
subgroups of G, where N is a natural number. This is intimately related to the
fact that the metrics m can be regarded as N-parameter deformations of a
Riemannian metric in G/H with special symmetries. We show that there exists a
wealth of metrics having the aforementioned type of geodesics; those metrics
are constructed from any Lie subgroup series of the form H < H_1 < ... <
H_{N-1} < G.
