Extendable symplectic structures and the inverse problem of the calculus of variations for systems of equations written in an extended Kovalevskaya form

The talk is devoted to extendable symplectic structures for systems of equations written in an extended Kovalevskaya form.
It is shown, that each extension of a symplectic structure to jets is related to an extension of a special form.
Complete description of all extendable symplectic structures is obtained. Relation of this result with the inverse problem of the calculus of variations is discussed.
It is shown, that each variational formulation for a system of evolution equations is related to a two-sided invertible variational operator of a special form.