  We theoretically investigate dynamics of classical spins exchange-coupled
through an isotropic medium. The coupling is treated at the adiabatic level of
the medium's response, which mediates a first-order in frequency dissipative
interaction along with an instantaneous Heisenberg exchange. The resultant
damped spin precession yields exceptional points (EPs) in the coupled spin
dynamics, which should be experimentally accessible with the existing magnetic
heterostructures. In particular, we show that an EP is naturally approached in
an antiferromagnetic dimer by controlling local damping, while the same is
achieved by tuning the dissipative coupling between spins in the ferromagnetic
case. Extending our treatment to one-dimensional spin chains, we show how EPs
can emerge within the magnonic Brillouin zone by tuning the dissipative
properties. The critical point, at which an EP pair emerges out of the
Brillouin zone center, realizes a gapless Weyl point in the magnon spectrum.
Tuning damping beyond this critical point produces synchronization (level
attraction) of magnon modes over a finite range of momenta, both in ferro- and
antiferromagnetic cases. We thus establish that damped magnons can generically
yield singular points in their band structure, close to which their kinematic
properties, such as group velocity, become extremely sensitive to the control
parameters.
