  We construct complete normal forms for $5$-dimensional real hypersurfaces in
$\mathbb C^3$ which are $2$-nondegenerate and also of Levi non-uniform rank
zero at the origin point ${\bf p} =0$. The latter condition means that the rank
of the Levi form vanishes at ${\bf p}$ but not identically in a neighborhood of
it. The mentioned hypersurfaces are the only finitely nondegenerate real
hypersurfaces in $\mathbb C^3$ for which their complete normal forms were
absent in the literature. As a byproduct, we also treat the underlying
biholomorphic equivalence problem between the hypersurfaces. Our primary
approach in constructing the desired complete normal forms is to utilize the
techniques derived in the theory of equivariant moving frames. It notably
offers the advantage of systematic and symbolic manipulation of the associated
computations.
