In this paper, we consider real hypersurfaces $M$ in $\mathbb
{C}^3$ (or, more generally, $5$-dimensional CR (Cauchy-Riemann)
manifolds of hypersurface type) at uniformly Levi degenerate points,
that is, Levi degenerate points such that the rank of the Levi form is
constant in a neighborhood. We also require that the hypersurface
satisfy a certain second-order nondegeneracy condition (called
$2$-nondegeneracy) at the point. One of our main results is the
construction, near any point $p_0\in M$ satisfying the above
conditions, of a principal bundle $P\to M$ and a $\mathbb {R}^{\dim
P}$-valued 1-form $\underline {\omega}$, uniquely determined by the
CR structure on $M$, which defines an absolute parallelism on $P$. If
$M$ is real-analytic, then covariant derivatives of $\underline
{\omega}$ yield a complete set of local biholomorphic invariants for
$M$. This solves the biholomorphic equivalence problem for uniformly
Levi degenerate hypersurfaces in $\mathbb {C}^3$ at
$2$-nondegenerate points.