  The boundary integral equation method ascertains explicit relations between
localized surface phonon and plasmon polariton resonances and the eigenvalues
of its associated electrostatic operator. We show that group-theoretical
analysis of Laplace equation can be used to calculate the full set of
eigenvalues and eigenfunctions of the electrostatic operator for shapes and
shells described by separable coordinate systems. These results not only unify
and generalize many existing studies but also offer the opportunity to expand
the study of phenomena like cloaking by anomalous localized resonance. For that
reason we calculate the eigenvalues and eigenfunctions of elliptic and circular
cylinders. We illustrate the benefits of using the boundary integral equation
method to interpret recent experiments involving localized surface phonon
polariton resonances and the size scaling of plasmon resonances in graphene
nano-disks. Finally, symmetry-based operator analysis can be extended from
electrostatic to full-wave regime. Thus, bound states of light in the continuum
can be studied for shapes beyond spherical configurations.
