  In this paper we prove new embedding results for compactly supported
deformations of $CR$ submanifolds of $\mathbb{C}^{n+d}$: We show that if $M$ is
a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$,
then any compactly supported $CR$ deformation stays in the space of globally
$CR$ embeddable in $\mathbb{C}^{n+d}$ manifolds. This improves an earlier
result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$
submanifold of $\mathbb{C}^{n+d}$. We also give examples of weakly
$2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$
deformations that are not even locally $CR$ embeddable.

Complex Variables;Analysis of PDEs;Differential Geometry