In mathematics, any Lagrangian system generally admits gauge symmetries, though it may happen that they are trivial.
In theoretical physics, the notion of gauge symmetries depending on parameter functions is a cornerstone of contemporary field theory.
taking its values in the linear space of (variational or exact) symmetries of
[1] For instance, this is the case of gauge symmetries in classical field theory.
Note that, in quantum field theory, a generating functional may fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts.