  Exactly solvable models play a special role in Condensed Matter Physics,
serving as secure theoretical starting points for investigation of new
phenomena. Changlani et al. [Phys. Rev. Lett. 120, 117202 (2018)] have
discovered a limit of the XXZ model for S=1/2 spins on the kagome lattice,
which is not only exactly solvable, but features a huge degeneracy of exact
ground states corresponding to solutions of a three-coloring problem. This
special point of the model was proposed as a parent for multiple phases in the
wider phase diagram, including quantum spin liquids. Here, we show that the
construction of Changlani et al. can be extended to more general forms of
anisotropic exchange interaction, finding a line of parameter space in an XYZ
model which maintains both the macroscopic degeneracy and the three-coloring
structure of solutions. We show that the ground states along this line are
partially ordered, in the sense that infinite-range correlations of some spin
components coexist with a macroscopic number of undetermined degrees of
freedom. We therefore propose the exactly solvable limit of the XYZ model on
corner-sharing triangle-based lattices as a tractable starting point for
discovery of quantum spin systems which mix ordered and spin liquid-like
properties.
