  A conservation law theorem stated by N. Ibragimov along with its subsequent
extensions are shown to be a special case of a standard formula that uses a
pair consisting of a symmetry and an adjoint-symmetry to produce a conservation
law through a well-known Fr\'echet derivative identity. Also, the connection of
this formula (and of Ibragimov's theorem) to the standard action of symmetries
on conservation laws is explained, which accounts for a number of major
drawbacks that have appeared in recent work using the formula to generate
conservation laws. In particular, the formula can generate trivial conservation
laws and does not always yield all non-trivial conservation laws unless the
symmetry action on the set of these conservation laws is transitive. It is
emphasized that all local conservation laws for any given system of
differential equations can be found instead by a general method using
adjoint-symmetries. This general method is a kind of adjoint version of the
standard Lie method to find all local symmetries and is completely algorithmic.
The relationship between this method, Noether's theorem, and the
symmetry/adjoint-symmetry formula is discussed.
