Almost every path structure is not variational

Given a smooth family of unparameterized curves such that through every point in direction there passes exactly one curve, does exist Lagrangian with extremals being precisely this family? It is known dimension 2 the answer positive. In 3, it follows from work Douglas is, general, negative. We generalise result to all higher dimensions and show actually negative for almost curves, also as path structure or geometry. On other hand, we consider geometries possessing infinitesimal symmetries projective structures submaximal symmetry are variational. Note algebra, so-called Egorov structure, not pseudo-Riemannian metrizable; metrizable class Kropina pseudo-metrics explicitly construct corresponding Lagrangian.