How can one compute the wave cone $\Lambda_{\mathcal A}$, defined as \begin{equation*} \Lambda_{\mathcal A}:=\bigcup_{|\xi|=1} \ker \mathbb A^k(\xi) \qquad\textrm{with}\qquad \mathbb A^k(\xi)= (2\pi i)^{k} \sum_{|\alpha|\le k}A_{\alpha}\xi^{\alpha}, \end{equation*}
of the (row-wise) curl operator such that for every $M \in \mathbb R^{M \times N}$, $$\mathcal A (M) = \operatorname{curl} (M).$$
The result is
$$\Lambda_{\mathcal A} = \bigcup_{\xi \in \S^{d-1}} \ker \mathbb A(\xi) = \left\{a \otimes \xi \in \mathbb R^{M \times N}:\, a \in \mathbb R^M,\, \xi \in \mathbb S^{N-1}\right\}. $$
How does one prove it?