It is shown that complexities in a problem of elastic scattering of a photon on a pair of Rayleigh particles (two small metallic spheres) are similar to the complexities of the classic problem of three bodies in celestial mechanics. In the latter problem, as is well known, the phase trajectory of a system becomes a nonanalytical function of its variables. In our problem, the trajectory of a virtual photon at some frequency could be considered such as the well-known Antoine set (Antoine's necklace) or a chain with interlaced sections having zero topological dimension and fractal structure. Such a virtual "zero-dimensional" photon could be localized between the particles of the pair. The topology suppresses the photon's exit to the real world with dimensional equal-to-or-greater-than units. The physical reason for this type of photon localization is related to the "mechanical rigidity" of interlaced sections of the photon trajectory due to a singularity of energy density along these sections. Within the approximations used in this paper, the effect is possible if the frequency of the incident radiation is equal to double the frequency of the dipole surface plasmon in an isolated particle, which is the only character frequency in the problem. This condition and transformation of the photon trajectory to the zero-dimensional Antoine set reminds of some of the simplest variants of Poincar駳 catastrophe in the dynamics of some nonintegrable systems. The influence of the localization on elastic light scattering by the pair is investigated.