If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $X$ admits a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ of $J$ vanishes.

Assume $D\subset X$ is a J-invariant symplectic divisor (i.e. real codimension two). Under what condition on $J$, locally around points of $D$ (on some small open set around every $p\in D$), there are holomorphic functions

$$ F\colon U \to \mathbb{C}, \quad \bar\partial F =dF+i dF\circ J=0,$$ such that $D\cap U = F^{-1}(0)$.

I guess that the iff condition is the integrability condition in the normal direction to $D$, i.e. $$ N_J(u,v)\in TD, ~~\forall~ u\in TX|_D,~v\in TD,$$ but I am not sure whether this trivialy follows from the work of Newlander-Nirenberg or not.