  For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of the
three-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Each
of these maps has image being contained in a five-dimensional complex-linear
space and is of degree at most two, or equivalent to one of the four maps into
the five-dimensional sphere classified by Faran. As a byproduct of our
classification we obtain new examples of rational maps of degree three which
are $(N-3)$-degenerate only along a proper real subvariety and are not
equivalent to polynomial maps. In particular, by changing the base point, it is
possible to construct new families of nondegenerate maps.
