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We study the constitutive set $\mathcal{K}$ arising from a $2\times 2$ systemof conservation laws in one space dimension, endowed with one entropy andentropy-flux pair. The convexity properties of the set $\mathcal{K}$ relate tothe well-posedness of the underlying system and the ability to constructsolutions via convex integration. Relating to the convexity of $\mathcal{K}$,in the particular case of the $p$-system, Lorent and Peng [Calc. Var. PartialDifferential Equations, 59(5):Paper No. 156, 36, 2020] show that $\mathcal{K}$does not contain $T_4$ configurations. Recently, Johansson and Tione [arXive-prints, page arXiv:2208.10979, August 2022] showed that $\mathcal{K}$ doesnot contain $T_5$ configurations.  In this paper, we provide a substantial generalization of these results,based on a careful analysis of the shock curves for a large class of $2\times2$ systems. In particular, we provide several sets of hypothesis on generalsystems which can be used to rule out the existence of $T_4$ configurations inthe constitutive set $\mathcal{K}$. In particular, our results show thenonexistence of $T_4$ configurations for every well-known $2\times 2$hyperbolic system of conservation laws which verifies the Liu entropycondition.

Nonexistence of $T_4$ configurations for hyperbolic systems and the Liu entropy condition