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

SIGMA 7 (2011), 038, 12 pages      arXiv:1101.5975      https://doi.org/10.3842/SIGMA.2011.038
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

### First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

Claudia Chanu a, Luca Degiovanni b and Giovanni Rastelli b
a) Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, via Cozzi 53, Italia
b) Formerly at Dipartimento di Matematica, Università di Torino, Torino, via Carlo Alberto 10, Italia

Received January 31, 2011, in final form April 03, 2011; Published online April 11, 2011; Theorem 1, Lemmas 1 and 2, Example 2 are corrected January 02, 2012

Abstract
We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n-dimensional Hamiltonians L. The Liouville integrability of L implies the (minimal) superintegrability of H. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with L is constant. As examples, the procedure is applied to one-dimensional L, including and improving earlier results, and to two and three-dimensional L, providing new superintegrable systems.

Key words: superintegrable Hamiltonian systems; polynomial first integrals; constant curvature; Hessian tensor.

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