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The theory of Flower Constellations consists of a general methodology to design axial-symmetric satellite constellations whose satellites all move on the same trajectory with respect to a reference frame rotating at assigned constant angular velocity. Due to the complexity, the Flower Constellations theory as introduced by the authors, is incomplete, leaving open several problems, such as the equivalency and the similarity. In this article, the foundations of the Flower Constellations theory are revisited from a mathematical perspective, and three new important invariants of these constellations are found constituting fundamentals toward the complete theory. These are: the -space, the flower anomaly, and the configuration number. Using these elements, we provide a theorem giving the necessary and sufficient conditions for the equivalency problem in the -space and an algorithm allowing to generate all the similar Flower Constellations to a given one.