  We present a wide class of partially integrable lattice models with two-spin
interactions, which generalize the Kitaev honeycomb model. These models have an
infinite number of conserved quantities associated with each plaquette of the
lattice, conserved large loop operators on the torus, and protected topological
degeneracy. We introduce a `slave-genon' approach, which generalizes the
Majorana fermion approach in the Kitaev honeycomb model. The Hilbert space of
our spin model can be embedded into an enlarged Hilbert space of non-Abelian
twist defects, referred to as genons. In the enlarged Hilbert space, the spin
model is exactly reformulated as a model of non-Abelian genons coupled to a
discrete gauge field. We discuss in detail a particular $Z_3$ generalization,
and show that in a certain limit the model is analytically tractable and may
produce a non-Abelian topological phase with chiral parafermion edge states.
