  We introduce an extension of the Kitaev honeycomb model by including
four-spin interactions that preserve the local gauge structure and hence the
integrability of the original model. The extended model has a rich phase
diagram containing five distinct vison crystals, as well as a symmetric
$\pi$-flux spin liquid with a Fermi surface of Majorana fermions and a sequence
of Lifshitz transitions. We discuss possible experimental signatures and, in
particular, present finite-temperature Monte Carlo calculations of the specific
heat and the static vison structure factor. We argue that our extended model
emerges naturally from generic perturbations to the Kitaev honeycomb model.
