obtained by superposition of simple waves connecting different constant states: from a given left state ${\mathbf{U}}_\text{L}$ to a given right state ${\mathbf{U}}_\text{R}$ via an intermediate state ${\mathbf{U}}_\text{M}$ that is connected to ${\mathbf{U}}_\text{L}$ by a 1-wave (i.e., a Lax-shock or rarefaction associated to the first characteristic field), and to ${\mathbf{U}}_\text{R}$ by a 2-wave (i.e., a contact discontinuity). Since the GARZ system is—as the ARZ model—of Temple class [@Temple1983], shocks and rarefaction wave curves in phase space coincide. Moreover, due to Lemma \[lem:lambdas\_decreasing\], a simple wave of the first family is either a shock or a rarefaction wave.

In the phase space $(\rho,w)$-plane we discuss the shape of the characteristic fields. The second fields are parallel to the $\rho$-axis, and the first fields are the contours $V(\rho,w) = \text{const}$. Since by assumption ${\frac{\partialV}{\partial\rho}}<0$ and ${\frac{\partialV}{\partialw}}>0$, the contours of $V(\rho,w)$ always have a finite and truly positive slope in the $(\rho,w)$-plane. Thus, for any two states $(\rho_\text{L},w_\text{L})$ and $(\rho_\text{R},w_\text{R})$ that satisfy $w_\text{L}\ge u_\text{R}$, where $u_\text{R} = V(\rho_\text{R},w_\text{R})$, there is a unique intermediate state $(\rho_\text{M},w_\text{M}) = (R(u_\text{R},w_\text{L}), w_\text{L})$, defined via the inverse function given in §\[subsubsec:regions\]. Moreover, because $\lambda^{(1)}$ is decreasing with $\rho$ (see Lemma \[lem:lambdas\_decreasing\]), the Lax entropy conditions [@Evans1998] imply that for 