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Let $M^{2n+1}$, $n\ge 1$, be a smooth manifold with a pseudo-convexintegrable CR structure of hypersurface type. We consider a sequence of CRinvariant subsets $ M=\mathcal S_0 \supset \mathcal S_1 \supset \cdots \supset\mathcal S_{n}, $ where $\mathcal S_q$ is the set of points where the Levi-formhas nullity $\ge q$. We prove that $\mathcal S_q$'s are locally given as commonzero sets of the coefficients $A_j,$ $j=0,1,\ldots, q-1,$ of the characteristicpolynomial of the Levi-form. Some sufficient conditions for local existence ofcomplex submanifolds are presented in terms of the coefficients $A_j$.

Nullity of the Levi-form and the associated subvarieties for pseudo-convex CR structures of hypersurface type