Nail Ibragimov - "Symmetries and conservation laws: a general theorem for arbitrary differential equations".

A general theorem on a connection between symmetries and conservation laws is proved for arbitrary differential equations. 
The theorem is valid also for any system of differential equations where the number of equations is equal to the number 
of dependent variables. The new theorem does not require existence of a Lagrangian and contains Noether’s theorem 
as a particular case. It is based on a concept of an adjoint equation for non-linear equations and the statement that the 
adjoint equation inherits all symmetries of the original equation proved recently by the author. Using the basic theorem, 
one can associate a conservation law with any group of Lie, Lie-Backlund  and nonlocal symmetries. 
The theorem is illustrated by computing conservation laws for several differential equations without classical Lagrangians. 
In particular, the Korteweg-de Vries equation provides a remarkable example on this theorem. Namely, it will be shown 
in the talk that all local symmetries of the KdV equation lead to trivial local conservation laws whereas its 
non-local symmetries provide all known local conservation laws.