Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Variety of Integrable Killing Tensors on the 3-Sphere   Integrable Killing tensors are used to classify orthogonal coordinates in
which the classical Hamilton-Jacobi equation can be solved by a separation of
variables. We completely solve the Nijenhuis integrability conditions for
Killing tensors on the sphere $S^3$ and give a set of isometry invariants for
the integrability of a Killing tensor. We describe explicitly the space of
solutions as well as its quotient under isometries as projective varieties and
interpret their algebro-geometric properties in terms of Killing tensors.
Furthermore, we identify all St\"ackel systems in these varieties. This allows
us to recover the known list of separation coordinates on $S^3$ in a simple and
purely algebraic way. In particular, we prove that their moduli space is
homeomorphic to the associahedron $K_4$.

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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph