We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws, having small bounded variation, in one space dimension, must be functions of special bounded variation ( \(\textrm{SBV}\) ). For more than one equation, this \(\textrm{SBV}\) -regularity for non-degenerate fluxes is new also in the case of systems of conservation laws outside the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of conservation laws: in such case we generalize the \(\textrm{SBV}\) -like regularity of the eigenvalue functions of the Jacobian matrix of the flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measure, a localization argument from [14, 40], and observations in real analysis.
