  Abstract deformations of the CR structure of a compact strictly pseudoconvex
hypersurface $M$ in $\mathbb{C}^2$ are encoded by complex functions on $M$. In
sharp contrast with the higher dimensional case, the natural integrability
condition for $3$-dimensional CR structures is vacuous, and generic
deformations of a compact strictly pseudoconvex hypersurface $M\subseteq
\mathbb{C}^2$ are not embeddable even in $\mathbb{C}^N$ for any $N$. A
fundamental (and difficult) problem is to characterize when a complex function
on $M \subseteq \mathbb{C}^2$ gives rise to an actual deformation of $M$ inside
$\mathbb{C}^2$. In this paper we study the embeddability of families of
deformations of a given embedded CR $3$-manifold, and the structure of the
space of embeddable CR structures on $S^3$. We show that the space of
embeddable deformations of the standard CR $3$-sphere is a Frechet submanifold
of $C^{\infty}(S^3,\mathbb{C})$ near the origin. We establish a modified
version of the Cheng-Lee slice theorem in which we are able to characterize
precisely the embeddable deformations in the slice (in terms of spherical
harmonics). We also introduce a canonical family of embeddable deformations and
corresponding embeddings starting with any infinitesimally embeddable
deformation of the unit sphere in $\mathbb{C}^2$.
