  We study the constitutive set $\mathcal{K}$ arising from a $2\times 2$ system
of conservation laws in one space dimension, endowed with one entropy and
entropy-flux pair. The convexity properties of the set $\mathcal{K}$ relate to
the well-posedness of the underlying system and the ability to construct
solutions via convex integration. Relating to the convexity of $\mathcal{K}$,
in the particular case of the $p$-system, Lorent and Peng [Calc. Var. Partial
Differential Equations, 59(5):Paper No. 156, 36, 2020] show that $\mathcal{K}$
does not contain $T_4$ configurations. Recently, Johansson and Tione [arXiv
e-prints, page arXiv:2208.10979, August 2022] showed that $\mathcal{K}$ does
not 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 $2\times
2$ systems. In particular, we provide several sets of hypothesis on general
systems which can be used to rule out the existence of $T_4$ configurations in
the constitutive set $\mathcal{K}$. In particular, our results show the
nonexistence of $T_4$ configurations for every well-known $2\times 2$
hyperbolic system of conservation laws which verifies the Liu entropy
condition.

Analysis of PDEs