On regularity and rigidity of $2\times 2$ differential inclusions into non-elliptic curves

We study differential inclusions $Du\in \Pi$ in an open set $\Omega\subset\mathbb R^2$, where $\Pi\subset \mathbb R^{2\times 2}$ is a compact connected $C^2$ curve without rank-one connections, but non-elliptic: tangent lines to $\Pi$ may have rank-one connections, so that classical regularity and rigidity results do not apply. For a wide class of such curves $\Pi$, we show that $Du$ is locally Lipschitz outside a discrete set, and is rigidly characterized around each singularity. Moreover, in the partially elliptic case where at least one tangent line to $\Pi$ has no rank-one connections, or under some topological restrictions on the tangent bundle of $\Pi$, there are no singularities. This goes well beyond previously known particular cases related to Burgers' equation and to the Aviles-Giga functional. The key is the identification and appropriate use of a general underlying structure: an infinite family of conservation laws, called entropy productions in reference to the theory of scalar conservation laws.