Generalization of the inverse problem of variational calculus. A complete solution. Part 1

The talk will discuss a generalization of the inverse problem of variational calculus and its solution.
The generalization of the inverse problem of variational calculus for a regular system of differential equations $F=0$ is the problem of describing of all linear differential operators in total derivatives $A$ such that $A(F)=E(L)$ for some Lagrangian $L$, here $E$ is the Euler operator. Such operators $A$ are called "variational" for the system of equation under consideration. The problem of constructing such operators turns out to be equivalent to the problem of computing of conservation laws of special form for another system naturally related to the given one.
We will show that the variational operators of regular system can be described in terms of cohomology of a complex on a corresponding diffiety. This yields interesting corollaries that will be discussed during the talk.
All our proofs are constructive. We will show a particular algorithm for constructing all variational operators of some fixed order for a regular system of equations. The algorithm is based on the possibility to describe some terms of the $C$-spectral sequence in both forms and operators terms.