  We introduce a class of uniformly $2$-nondegenerate CR hypersurfaces in
$\mathbb{C}^N$, for $N>3$, having a rank $1$ Levi kernel. The class is first of
all remarkable by the fact that for every $N>3$ it forms an {\em explicit}
infinite-dimensional family of everywhere $2$-nondegenerate hypersurfaces. To
the best of our knowledge, this is the first such construction. Besides, the
class an infinite-dimensional family of non-equivalent structures having a
given constant nilpotent CR symbol for every such symbol. Using methods that
are able to handle all cases with $N>5$ simultaneously, we solve the
equivalence problem for the considered structures whose symbol is represented
by a single Jordan block, classify their algebras of infinitesimal symmetries,
and classify the locally homogeneous structures among them. We show that the
remaining considered structures, which have symbols represented by a direct sum
of Jordan blocks, can be constructed from the single block structures through
simple linking and extension processes.
