We consider the quasi-static problem governing the localized surface plasmon modes and permittivityeigenvaluesεof smooth, arbitrarily shaped, axisymmetric inclusions. We develop an asymptotic theoryfor the dense part of the spectrum, i.e., close to the accumulation valueε=−1 at which a flat interfacesupports surface plasmons; in this regime, the field oscillates rapidly along the surface and decays expo-nentially away from it on a comparable scale. Withτ=−(ε+1)as the small parameter, we developa surface-ray description of the eigenfunctions in a narrow boundary layer about the interface; the fastphase variation, as well as the slowly varying amplitude and geometric phase, along the rays are deter-mined as functions of the local geometry. We focus on modes varying at most moderately in the azimuthaldirection, in which case the surface rays are meridian arcs that focus at the two poles. Asymptoticallymatching the diverging ray solutions with expansions valid in inner regions in the vicinities of the polesyields the quantization rule1τ∼πnΘ+12(πΘ−1)+o(1),wheren 1 is an integer andΘa geometric parameter given by the product of the inclusion length andthe reciprocal average of its cross-sectional radius along its symmetry axis. For a sphere,Θ=π, wherebythe formula returns the exact eigenvaluesε=−1−1/n. We also demonstrate good agreement with exactsolutions in the case of prolate spheroids