Geodesic
Remarks on gauge invariance and first-class constraints
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of modern theoretical and mathematical physics: Gauge theories and superstrings, Tome 272 (2011), pp. 152-161.

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Gauge symmetries lead to first-class constraints. This assertion is, of course, true only for non-trivial gauge symmetries, i.e., gauge symmetries that act non-trivially on-shell on the dynamical variables. We illustrate this well-appreciated fact for time reparametrization invariance in the context of modifications of gravity – suggested in a recent proposal by Hořava – in which the Hamiltonian constraint is deformed by arbitrary spatial diffeomorphism invariant terms, where some subtleties are found to arise. 
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Marc Henneaux; Axel Kleinschmidt; Gustavo Lucena Gómez. Remarks on gauge invariance and first-class constraints. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of modern theoretical and mathematical physics: Gauge theories and superstrings, Tome 272 (2011), pp. 152-161. http://geodesic.mathdoc.fr/item/TM_2011_272_a11/

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