Cone structures on differentiable manifolds are defined as fields of cones in the tangent bundle. A cone structure is isotrivial, modeled on a given projective variety, if the cones at each point of the manifold are projectively equivalent to that variety. We study integrable cone structures, which arise naturally from ordinary differential equations via a canonical construction. We prove that isotrivial, integrable cone structures modeled on generic curves in $n$-dimensional projective space or on ruled surfaces in three-dimensional projective space are necessarily flat. We provide applications of the results to integrable systems and causal geometry. The talk is based on joint work with Tymon Frelik.