  Consider two hyperbolic systems of conservation laws in one space dimension
with the same eigenvalues and (right) eigenvectors. We prove that solutions to
Cauchy problems with the same initial data differ at third order in the total
variation of the initial datum. As a first application, relying on the
classical Glimm-Lax result, we obtain estimates improving those in by Saint
Raymond on the distance between solutions to the isentropic and non-isentropic
inviscid compressible Euler equations, under general equations of state.
Further applications are to the general scalar case, where rather precise
estimates are obtained, to an approximation by Di Perna of the p-system and to
a traffic model.
