  We consider the scattering of electromagnetic waves by non-spherical
dielectric resonators and reveal that it can be linked to the exceptional
points underpinned by the physics of non-Hermitian systems. We demonstrate how
symmetry breaking in the shape of an isolated dielectric nanoparticle can be
associated with the existence of an exceptional point in the eigenvalue
spectrum and formulate the general conditions for the strong coupling of
resonances, illustrating them for the example of the electric dipole and
magnetic dipole modes supported by a silicon nanoparticle. We argue that any
two modes of a dielectric nanoparticle can lead to an exceptional point
provided their resonant frequencies cross as a function of a tuning parameter,
such as, e.g. its aspect ratio, and their field distributions should have
opposite signs after a reflection in the transverse plane of the structure. The
coupled modes radiate as a mixture of electric and magnetic dipoles, which
result in a strong magnetoelectric response, being easily controlled by the
symmetry breaking perturbation. We also investigate the influence of a
dielectric substrate, demonstrating how the latter provides an additional
mechanism to tune the position of exceptional points in the parameter space.
Finally, we discuss applications of magnetoelectric exceptional points for
refractive index sensing.
