The structure of a narrow ray of gamma radiation propagating through a medium filled with free electrons (called Compton scatterer) is investigated. The dual representation is used including primary (basic) and adjoint forms of kinetic equations. In the ray propagation problem primary form possesses cylindrical symmetry whereas the adjoint form (in case of isotropic detector in an unbounded homogeneous medium) has the spherical symmetry. The analysis performed on the base of adjoint function

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... gularities shows, in particular, that in a vicinity of the narrow gamma-ray singlescattered radiation predominates over all other components. Analytical representation of the field in the vicinity of a cylindrical primary ray has been found. The result can be important in gamma astronomy processing.