  We prove the existence of generalized characteristics for weak, not
necessarily entropic, solutions of Burgers' equation \[ \partial_t u
+\partial_x \frac{u^2}{2} =0, \] 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.
