  In this paper, we introduce the geodesic orbit and weakly symmetric
properties in homogeneous spray geometry. When a homogeneous spray manifold is
endowed with a reductive decomposition, we can use the spray vector field to
describe these properties, and prove that a weakly symmetric spray manifold
must be geodesic orbit, which generalizes its analog in homogeneous Riemannian
and Finsler geometries.
