  Topological insulators are most frequently constructed using lattices with
specific degeneracies in their linear spectra, such as Dirac points. For a
broad class of lattices, such as honeycomb ones, these points and associated
Dirac cones generally appear in non-equivalent pairs. Simultaneous breakup of
the time-reversal and inversion symmetry in systems based on such lattices may
result in the formation of the unpaired Dirac cones in bulk spectrum, but the
existence of topologically protected edge states in such structures remains an
open problem. Here photonic Floquet insulator on honeycomb lattice with
unpaired Dirac cones in its spectrum is introduced that can support
unidirectional edge states appearing at the edge between two regions with
opposite sublattice detuning. Topological properties of this system are
characterized by the nonzero valley Chern number. Remarkably, edge states in
this system can circumvent sharp corners without inter-valley scattering even
though there is no total forbidden gap in the spectrum. Our results reveal
unusual interplay between two different physical mechanisms of creation of
topological edge states based on simultaneous breakup of different symmetries
of the system.
