  In this paper we present an analytical solution for the eigenmodes and
corresponding electric field of a composite system made of a nanorod in the
vicinity of a plasmonic semi-infinite metallic system. To be specific, we
choose Silver as the material for both the nanorod and the semi-infinite metal.
The system is composed of two sub-systems with different symmetries: the rod
has symmetry, while the interface has a rectangular one. Using a boundary
integral method, proposed by Eyges, we are able to compute analytically the
integrals that sew together the two systems. In the end, the problem is reduced
to one of linear algebra, where all the terms in the system are known
analytically. For large distances between the rod and the planar surface, only
a few of those integrals are needed and a full analytical solution can be
obtained. Our results are important to benchmark other numerical approaches and
represent a starting point in the discussion of systems composed of nanorods
and two-dimensional materials.
