  We make significant progress toward the classification of 2nd order
superintegrable systems on 3-dimensional conformally flat space that have
functionally linearly dependent (FLD) symmetry generators, with special
emphasis on complex Euclidean space. The symmetries for these systems are
linearly dependent only when the coefficients are allowed to depend on the
spatial coordinates. The Calogero-Moser system with 3 bodies on a line and
2-parameter rational potential is the best known example of an FLD
superintegrable system. We work out the structure theory for these FLD systems
on 3D conformally flat space and show, for example, that they always admit a
1st order symmetry. A partial classification of FLD systems on complex 3D
Euclidean space is given. This is part of a project to classify all 3D 2nd
order superintegrable systems on conformally flat spaces.
