Given an almost complex structure, the Nijenhuis tensor is the obstruction to the existence of pseudo-holomorphic functions. If it vanishes identically there exists maximal number of independent pseudo-holomorphic functions, which is the case of the Newlander-Nirenberg theorem. We discuss the partial integrablity depending on the rank of the Nijenhuis tensor and the existence of a pseudo-holomorphic function on the zero locus. As an application we discuss the solvability of quasi-linear Cauchy-Riemann equations.