We study the tangential Cauchy-Riemann equations ${\overline{\partial }}_{b}u=\omega$ for $\left(0,q\right)$-forms on quadratic $CR$ manifolds. We discuss solvability for data $\omega$ in the Schwartz class and describe the range of the tangential Cauchy-Riemann operator in terms of the signatures of the scalar components of the Levi form.

For pseudocomplex abstract $CR$ manifolds, the validity of the Poincaré Lemma for $\left(0,1\right)$ forms implies local embeddability in ${\mathbb{C}}^N$. The two properties are equivalent for hypersurfaces of real dimension $\ge 5$. As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for $\left(0,1\right)$ forms for a large class of abstract $CR$ manifolds of $CR$ codimension larger than one.