  We study solutions of the first order partial differential inclusions of the
form $\nabla u\in K$, where $u:\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$ and
$K$ is a set of $m\times n$ real matrices, and derive a companion version to
the result of {M\"uller and \v{S}ver\'ak} [20], concerning a general linear
constraint on the components of $\nabla u$. We then consider two applications:
the vectorial eikonal equation and a $T_4$-configuration both under linear
constraints.
