Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations.


How about relations among relations among the Euler–Lagrange equations, cf. syzygies?



higher order Noether identities
Identities among Noether identities (lets call them *higher stage* Noether identities, for lack of a better standard name) are mapped to gauge symmetries of gauge symmetries (lets call them *higher stage* gauge symmetries, also sometimes known as *reducibility identities*). As far as I know, references to the notions of higher stage Noether identities and higher stage gauge symmetries and to this result are scattered through, and sometimes hidden in, the literature on the BRST treatment of gauge theories. For instance, the construction of the Koszul-Tate differential in the Batalin-Vilkovisky (BV) treatment of gauge theories requires that each generation of ghosts (ghosts of ghost, ghosts of ghosts of ghosts, etc., which generate higher stage gauge symmetries) has corresponding anti-fields (which generate higher gauge Noether identities). I'm sure this is well known to you, Jim, since this well known review article by Marc Henneaux (Secs. 2.10, 5.4) mentions it and refers to one of your old articles for details:


* Henneaux, M. Lectures on the antifield-BRST formalism for gauge theories. Nuclear Physics B - Proceedings Supplements **18**, 47-105 (1990). <http://dx.doi.org/10.1016/0920-5632(90)90647-d>.
* Fish, J., Henneaux, M., Stasheff, J. Comm. Math. Phys. **120**, 379 (1989).


A more recent reference that mentions the identification between higher stage gauge symmetries and higher stage Noether identities is Sec. 3.2 of


* Peter O. Kazinski, Simon L. Lyakhovich and Alexey A. Sharapov. Lagrange structure and quantization. JHEP **07** (2005)076 [doi:10.1088/1126-6708/2005/07/076](http://dx.doi.org/10.1088/1126-6708/2005/07/076) [arXiv:hep-th/0506093](http://arxiv.org/abs/hep-th/0506093)


There is also a generalization of Noether's first theorem which puts higher stage Noether identities into correspondence with higher degree conserved currents (where the latter refers to on-shell closed differential forms of lower form-degree). I can add references for that as well, if it is part of your question.


