  The present article provides a study of $2-$Killing vector fields on warped
product manifolds as well as characterization of this structure on standard
static and generalized Robertson-Walker space-times. Some conditions for a
$2-$Killing vector field on a warped product manifold to be parallel are
obtained. Moreover, some results on the curvature of a warped product manifolds
in terms of $2-$Killing vector fields are derived. Finally, we apply our
results to describe $2-$Killing vector fields of some well known warped product
space-time models.
