  It is known that the set of all solutions of a commutant lifting and other
interpolation problems admits a Redheffer linear-fractional parametrization.
The method of unitary coupling identifies solutions of the lifting problem with
minimal unitary extensions of a partially defined isometry constructed
explicitly from the problem data. A special role is played by a particular
unitary extension, called the central or universal unitary extension. The
coefficient matrix for the Redheffer linear-fractional map has a simple
expression in terms of the universal unitary extension. The universal unitary
extension can be seen as a unitary coupling of four unitary operators (two
bilateral shift operators together with two unitary operators coming from the
problem data) which has special geometric structure. We use this special
geometric structure to obtain an inverse theorem (Theorem 8.4 as well as
Theorem 9.3) which characterizes the coefficient matrices for a Redheffer
linear-fractional map arising in this way from a lifting problem. When
expressed in terms of Hellinger-space functional models (Theorem 10.3), these
results lead to generalizations of classical results of Arov and to
characterizations of the coefficient matrix-measures of the lifting problem in
terms of the density properties of the corresponding model spaces. The main
tool is the formalism of unitary scattering systems developed in [18], [45].
