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For $N \geq 4$ we classify the $(N-3)$-degenerate smooth CR maps of thethree-dimensional unit sphere into the $(2N-1)$-dimensional unit sphere. Eachof these maps has image being contained in a five-dimensional complex-linearspace and is of degree at most two, or equivalent to one of the four maps intothe five-dimensional sphere classified by Faran. As a byproduct of ourclassification we obtain new examples of rational maps of degree three whichare $(N-3)$-degenerate only along a proper real subvariety and are notequivalent to polynomial maps. In particular, by changing the base point, it ispossible to construct new families of nondegenerate maps.

On highly degenerate CR maps of spheres