An LED optical element is proposed as an alternative to cold-cathode fluorescent lamps. The optical element generates two symmetric uniformly illuminated line segments on the diffuse reflector. The illuminated segments then act as secondary linear light sources. The calculation of the optical element is reduced to the integration of the system of two explicit ordinary differential equations. The results of the simulation of an illumination system module consisting of a set of optical elements generating a set of line segments on the surface of the diffuse reflector are presented. The elements are located directly on the surface of the reflector. The simulation results demonstrate the uniform illumination of a rectangular area at a distance of 30-40 mm from the light source plane. The lighting efficiency of the designed system exceeds 83%.