  We study theoretically and numerically high density of states for hyperbolic
bilayered metamaterials (HMM). It reveals that density response of HMM is
reminiscent of Fermi electronic band structure of metal or semiconductors.
  By the method of Green function a van Hove type singularity is found in
photonic density spectra of HMM with saddle point localization on photonic
Fermi surface (FS) of metamaterial. Similar to the electronic systems, the
photonic FS experiences instabilities induced by the changes in volume
fractions of its constituents that leads to the Lifshitz type zero-temperature
phase transition between FS of types I and II hyperbolic states at the topology
protected critical point.
