  This article constructs a surface whose Neumann-Poincar\'e (NP) integral
operator has infinitely many eigenvalues embedded in its essential spectrum.
The surface is a sphere perturbed by smoothly attaching a conical singularity,
which imparts essential spectrum. Rotational symmetry allows a decomposition of
the operator into Fourier components. Eigenvalues of infinitely many Fourier
components are constructed so that they lie within the essential spectrum of
other Fourier components and thus within the essential spectrum of the full NP
operator. The proof requires the perturbation to be sufficiently small, with
controlled curvature, and the conical singularity to be sufficiently flat.
