function constructions, as discussed in [@scalar_move_entire_solution].

In this paper, we bring novel ideas from the scalar case in [@scalar_move_entire_solution] to the systems case. In the systems case, we need to use the theory of a-contraction.

For the case of scalar, the generalized characteristics for $u$ are the natural shift functions to be using. In the systems case, we use a shift function which again is based on the generalized characteristics, but with a correction where the shift travels at greater-than-characteristic-speed due to a-contraction and the existence of multiple shock families in the systems case.

On top of the benefits for generalized-characteristic-based shifts mentioned in [@scalar_move_entire_solution] (such as simplicity of analysis, ease of construction, enhanced control on the shifts, and strictly negative entropy creation) the use of generalized-characteristic-based shifts for the *systems case* allows for simplified proofs compared to the previous state-of-the-art a-contraction result, [@MR3519973]. By having very obvious control on the speed of generalized-characteristic-based shifts, we are able to obviate the need for many of the computations in the foregoing analysis [@MR3519973].

For systems of conservation laws in one space dimension such as (including the scalar conservation laws), we have non-uniqueness for solutions. We impose entropy conditions such as , 