Gauge symmetry of Lagrangian is defined as the differential operator on some vector bundle with the condition of its value taken in linear space.
Gauge symmetry is depends on vector bundle and its partial derivatives. Example of such types of dependencies in Yang-Mills theory, classical field theory and Gauge gravitation theory.
There happens some odd things in gauge symmetry
Being Lagrangian symmetry, gauge symmetry satisfies first Noethers theorem.
According to second Noether theorem, Gauge symmetry and Noether identities have a relationship of one to one correspondence. This one to one correspondence is satisfies by Euler-Lagrange operator.
Gauge symmetry characterize the degeneracy of Lagrangian system.
In terms of quantum field theory, there found another complexity. There a generating functional fail may found invariant under gauge transformation. In this case gauge symmetry is replaced and BRST symmetry is introduces instead of it.
