Say we have a 4 dimensional real manifold and two 2-forms, called $k$ and $\omega$. Assume $$ d k = 0, \hspace{1 cm} d \omega = 0. $$ Furthermore, assume $\omega$ is non-degenderate, but $k$ is not. In particular, $k$ is rank-2, meaning there is a 2-dimensional vector space on which $k$ vanishes. Finally, assume $$ k \wedge \omega = 0. $$
My question is, is it possible to always find a set of local coordinates $(q_1, q_2, p_1, p_2)$ such that we simultaneously have, $$ \omega = d q_1 \wedge d p_1 + d q_2 \wedge d p_2 $$ $$ k = d q_1 \wedge d q_2. $$
Of course, if $k$ was not in the picture, this would just be Darboux's theorem. But here I am asking for a stronger statement.