  It is known that pure row contractions with one-dimensional defect spaces can
be classified up to unitary equivalence by compressions of the standard
$d$-shift acting on the full Fock space. Upon settling for a softer relation
than unitary equivalence, we relax the defect condition and simply require the
row contraction to admit a cyclic vector. We show that cyclic pure row
contractions can be "transformed" (in a precise technical sense) into
compressions of the standard $d$-shift. Cyclic decompositions of the underlying
Hilbert spaces are the natural tool to extend this fact to higher
multiplicities. We show that such decompositions face multivariate obstacles of
an algebraic nature. Nevertheless, some decompositions are obtained for
nilpotent commuting row contractions by analyzing function theoretic rigidity
properties of their invariant subspaces.
