Classification of Symmetric Four-Body Dziobek Central Configurations and Application to the Earth--Moon System
Central configurations are fundamental equilibrium solutions of the Newtonian $n$-body problem and play a key role in understanding the structure and dynamics of gravitational systems. However, the classification and enumeration of such configurations remain incomplete in the four-body case, particularly for symmetric configurations. In this work, we develop a framework for determining and classifying symmetric four-body Dziobek configurations. The method allows the explicit determination of the number of admissible configurations directly from the mass parameters, without requiring prior knowledge of their geometric structure. Combined with previously established semi-analytical relations, this approach provides a systematic characterization of symmetric configurations in terms of mass ratios. As a physically relevant application, we apply the framework to the Earth--Moon system and determine the possible symmetric four-body central configurations involving Earth- and Moon-mass bodies and an additional object of arbitrary mass. We identify both isolated configurations and continuous families of equilibrium solutions, extending the concept of libration points to the four-body problem. The presented semi-analytical approach contributes to the understanding of equilibrium structures in multi-body gravitational systems and provides a foundation for further studies in celestial mechanics, planetary dynamics, and spacecraft motion in complex gravitational environments.