  Beginning from the conventional square-lattice nearest-neighbor
antiferromagnetic Heisenberg model, we allow the $J_x$ and $J_y$ couplings to
be anisotropic, with their values depending on the bond orientation. The
emergence of anisotropic, bond-dependent, couplings should be expected to occur
naturally in most antiferromagnetic compounds which undergo structural
transitions that reduce the point-group symmetry at lower temperature. Using
the spin-wave approximation, we study the model in several parameter regimes by
diagonalizing the reduced Hamiltonian exactly, and computing the edge spectrum
and Berry connection vector, which show clear evidence of localized topological
charges. We discover phases that exhibit Weyl-type spin-wave dispersion,
characterized by pairs of degenerate points and edge states, as well as phases
supporting lines of degeneracy. We also identify a parameter regime in which
there is an exotic state hosting gapless linear spin-wave dispersions with
different longitudinal and transverse spin-wave velocities.
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