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 $D\cap U = F^{-1}(0)$.

** A weaker, and may be more feasible question is to ask for the $J$-holomorphicity only along $D$.