We prove that for a real analytic generic submanifold $M$ of ${\mathbb{C}}^n$ whose Levi-form has constant rank, the tangential ${\overline{\partial }}_M$-system is non-solvable in degrees equal to the numbers of positive and $M$ negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with $M$ non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.