Abstract:
We consider a very special class of differential equations, which is characterized by the condition that all its local differential invariants (under the action of a suitable Lie pseudogroup) become first integrals when restricted to the equation manifold. Such differential equations were introduced in a short note of Elie Cartan (Les espaces généralisés et l'intégration de certaines classes d'équations différentielles, C.R., 1938, V.206, N.23, 1689-1693), who characterized them in two simplest cases: scalar 2nd order ODEs viewed under the pseudogroup of point transformations and scalar 3rd order ODEs under the group of contact transformations. We show how these results generalize to any systems of ODEs and, more generally, differential equations of finite type. The same question for arbitrary systems of PDEs still remains open.
