A
general theorem on conservation laws for arbitrary differential
equations is proved. 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 is based on a concept of an adjoint
equation for non-linear equations suggested recently by the author. It
is proved that the adjoint equation inherits all symmetries of the
original equation. Accordingly, one can associate a conservation law
with any group of Lie, Lie–Bäcklund or non-local symmetries and find
conservation laws for differential equations without classical
Lagrangians.
