Paraconformal structures and differential equations

In the present paper we consider manifolds equipped with a paraconformal structure, understood as the tangent bundle isomorphic to a symmetric tensor product of rank-two vector bundles. If an ordinary differential equation satisfies Wünschmann condition then it defines a paraconformal structure on solution space. In the present paper we characterize all paraconformal structures which can be obtained in this way. In particular, we provide a new proof that all paraconformal structures on 3-dimensional manifolds are defined by ODEs. We show that if the dimension is greater than 3 then there exist structures which are not defined by an ODE.