  We interpret the property of having an infinitesimal symmetry as a
variational property in certain geometric structures. This is achieved by
establishing a one-to-one correspondence between a class of cone structures
with an infinitesimal symmetry and geometric structures arising from certain
systems of ODEs that are variational. Such cone structures include
pseudo-Riemannian conformal structures and distributions of growth vector
(2,3,5) and (3,6). The correspondence is obtained via symmetry reduction and
quasi-contactification. Subsequently, for each class of such cone structures we
provide invariant conditions that imply more specific properties, such as
having a null infinitesimal symmetry, being foliated by null submanifolds, or
having reduced holonomy to the appropriate contact parabolic subgroup. As an
application of our results we give an alternative proof of the variationality
of chains in CR geometry.
