  A unitarily invariant, complete Nevanlinna--Pick kernel $K$ on the unit ball
determines a class of operators on Hilbert space called $K$-contractions. We
study those $K$-contractions that are constrained, in the sense that they are
annihilated by an ideal of multipliers. Our overarching goal is to identify
various joint spectra of these constrained $K$-contractions through the
vanishing locus of their annihilators. Our methods are based around a careful
analysis of a subset of the ball associated to the annihilator, which we call
its support. For the functional models, we show how this support completely
determines several natural joint spectra. The picture is more complicated for
general $K$-contractions, as their spectra can be properly contained in the
support. Nevertheless, the "localizable" portion of the support always consists
of spectral points. When the support is assumed to be small in an appropriate
sense, we manage to effectively detect points of localizability.
