  The aim of this work is to extend and prove the Onsager conjecture for a
class of conservation laws that possess generalized entropy. One of the main
findings of this work is the "universality" of the Onsager exponent, $\alpha >
1/3$, concerning the regularity of the solutions, say in $C^{0,\alpha}$, that
guarantees the conservation of the generalized entropy; regardless of the
structure of the genuine nonlinearity in the underlying system.
