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We make significant progress toward the classification of 2nd ordersuperintegrable systems on 3-dimensional conformally flat space that havefunctionally linearly dependent (FLD) symmetry generators, with specialemphasis on complex Euclidean space. The symmetries for these systems arelinearly dependent only when the coefficients are allowed to depend on thespatial coordinates. The Calogero-Moser system with 3 bodies on a line and2-parameter rational potential is the best known example of an FLDsuperintegrable system. We work out the structure theory for these FLD systemson 3D conformally flat space and show, for example, that they always admit a1st order symmetry. A partial classification of FLD systems on complex 3DEuclidean space is given. This is part of a project to classify all 3D 2ndorder superintegrable systems on conformally flat spaces.

symmetry, integrability and geometry: methods and applications

Toward Classification of 2nd Order Superintegrable Systems in  3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent  Symmetry Operators