Distributions associated to almost complex structures on symplectic manifolds

We look at methods to select triples (M,ω,J) consisting of a symplectic manifold (M,ω) endowed with a compatible positive almost complex structure J, in terms of the Nijenhuis tensor NJ associated to J. We study the image distribution ImNJ, which is the span at each point of the values of NJ, measuring the non integrability of J by the dimension of ImNJ.