Let $M \in \Bbb R^{3 \times 3}$. Let $\Omega\subset \mathbb{R}^3$ be a bounded Lipschitz domain and $u :\Omega \to \mathbb R$ be an infinitely-differentiable scalar field. What conditions should matrix $M$ satisfy to ensure the existence of a scalar field $v : \Omega \to \mathbb R$ such that
$$M\nabla u=\nabla v$$
In such a case, can we give an explicit expression of $v$ which depend on $M$ and $u$? Thanks.