The Antikythera mechanism is an ancient astronomical instrument with various functions to indicate the date of the Egyptian calendar, display the motions of celestial bodies, calculate astronomical periods, and predict eclipse events. However, the portions of the mechanism that have been excavated do not completely support its functions. In particular, the structures corresponding to the demonstration of solar and planetary motions have been lost. This paper presents a systematic reconstruction of design concepts for the lost structures with two degrees of freedom. According to the investigations of existing structure and other ancient astronomical instruments, design concepts with two degrees of freedom are generated, and three possible double-input conditions are identified. Based on a mechanism analysis, two types of reconstruction designs are developed: five-bar mechanisms with six joints and six-bar mechanisms with eight joints. Design constraints were determined in accordance with the historical literature and existing designs. Then, by applying the concepts of generalization and specialization, as well as the input conditions and manufacturing process, three, four, and seven feasible reconstruction designs are synthesized for the solar motion, inferior planetary motion, and superior planetary motion, respectively. In addition, relations for the teeth calculations are derived for detail designs.