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 conformal pseudo-Riemannian structures and distributions of growth vectors (2,3,5) and (3,6). In this talk we will primarily focus on conformal structures. The correspondence is obtained via symmetry reduction and quasi-contactification. Subsequently, we provide examples of each class of cone structures with 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, we show that chains in integrable CR structures of hypersurface type are metrizable. This is a joint work with Katja Sagerschnig.