  This paper describes and proves a canonical procedure to decouple
perturbations and optimize their gauge around backgrounds with one
non-homogeneous dimension, namely of co-homogeneity 1, while preserving
locality in this dimension. Derivatively-gauged fields are shown to have a
purely algebraic action; they can be decoupled from the other fields through
gauge-invariant re-definitions; a potential for the other fields is generated
in the process; in the remaining action each gauge function eliminates one
field without residual gauge. The procedure applies to spherically symmetric
and to cosmological backgrounds in either General Relativity or gauge theories.
The widely used ``gauge invariant perturbation theory'' is closely related. The
supplied general proof elucidates the algebraic mechanism behind it as well as
the method's domain of validity and its assumptions.
