### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 038, 17 pages      arXiv:1501.06601      https://doi.org/10.3842/SIGMA.2015.038

### Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

Joshua J. Capel a, Jonathan M. Kress a and Sarah Post b
a) Department of Mathematics, University of New South Wales, Sydney, Australia
b) Department of Mathematics, University of Hawai`i at Mānoa, Honolulu, HI, 96822, USA

Received February 03, 2015, in final form April 21, 2015; Published online May 08, 2015

Abstract
The invariant classification of superintegrable systems is reviewed and utilized to construct singular limits between the systems. It is shown, by construction, that all superintegrable systems on conformally flat, 3D complex Riemannian manifolds can be obtained from singular limits of a generic system on the sphere. By using the invariant classification, the limits are geometrically motivated in terms of transformations of roots of the classifying polynomials.

Key words: integrable systems; superintegrable systems; Lie algebra invariants; contractions.

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