We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers' equation & nbsp;& part;(t)u+& part;(x)u(2)/2=0,& nbsp;whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the L-2 stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution's hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws. (C)& nbsp;2022 The Author(s). Published by Elsevier Ltd.& nbsp;