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A synthesis method based on geometrical optics for designing a dual-reflector antenna system with an arbitrary phase and amplitude distribution in the aperture of the second reflector is presented. The first reflector may be illuminated by a pattern with an arbitrarily curved phase front. A pair of first-order ordinary nonlinear differential equations of the form are developed for the system. Questions concerning uniqueness, existence and bounds for the solutions can be answered. Calculations and numerical results for the design of a uniform amplitude and phase dual-reflector system are presented.