  We discuss the emergence and manipulation of generalised Dirac cones in the
subradiant collective modes of quantum metasurfaces. We consider a collection
of single quantum emitters arranged in a honeycomb lattice with subwavelength
periodicity. While conventional honeycomb quantum metasurfaces host bound modes
that display Dirac cones at the K and K' points, we show that introducing
uniaxial anisotropy in the lattice results in modified dispersion relations.
These include the tilting of Dirac cones, which changes the local density of
states at the Dirac point from vanishing (type I) to diverging (types II and
III), the emergence of semi-Dirac points, with linear and quadratic dispersions
in orthogonal directions, and the anisotropic movement of Dirac cones away from
the K and K' points. Such energy dispersions can modify substantially the
dynamics of local probes, such as quantum emitters, to which they have been
shown to induce anisotropic, power-law interactions
