Let (M, J ) be a germ of an almost complex manifold of real dimension 2m and let n (n < m) be an integer. We study a necessary and sufficient condition for M to admit an integrable submanifold N of complex dimension n. If n = m − 1, we find defining functions of N explicitly from the coefficients of the torsion tensor. For J obtained by small perturbation of the standard complex structure of C m this condition is given as an overdetermined system of second order PDEs on the perturbation. The proof is based on the rank conditions of the Nijenhuis tensor and application of the Newlander-Nirenberg theorem. We give examples of almost complex structures on C 3 : the ones with a single complex submanifold of dimension 2 and the ones with 1-parameter or 2-parameter families of complex submanifolds of dimension 2.
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