  The paper studies the locus in the rank 2 Higgs bundle moduli space
corresponding to points which are critical for d of the Poisson commuting
functions. These correspond to the Higgs field vanishing on a divisor of degree
D. The degree D critical locus has an induced integrable system related to
K(-D)-twisted Higgs bundles. Topological and differential-geometric properties
of the critical loci are addressed.
