Noether's theorem for diffeties

We will discuss how Noether's theorem for a system of Euler-Lagrange equations can be reformulated in terms of the corresponding diffiety. The possibility of such reformulating leads to the fact that the Noether's correspondence between symmetries and conservation laws can be lifted to coverings. After that I'll try to tell about a solution of the local inverse problem of the calculus of variations for regular systems of equations of the generalized Kovalevskaya form.