  We derive an integral equation describing surface-plasmon polaritons in
graphene deposited on a substrate with a planar surface and a dielectric
protrusion in the opposite surface of the dielectric slab. We show that the
problem is mathematically equivalent to the solution of a Fredholm equation,
which we solve exactly. In addition, we show that the dispersion relation of
the localized surface plasmons is determined by the geometric parameters of the
protrusion alone. We also show that such system supports both even and odd
modes. We give the electrostatic potential and the stream plot of the
electrostatic field, which clearly show the localized nature of the surface
plasmons in a continuous and flat graphene sheet.
