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

SIGMA 18 (2022), 039, 20 pages      arXiv:2112.01735      https://doi.org/10.3842/SIGMA.2022.039

### Doubly Exotic $N$th-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

İsmet Yurduşen a, Adrián Mauricio Escobar-Ruiz b and Irlanda Palma y Meza Montoya b
a) Department of Mathematics, Hacettepe University, 06800 Beytepe, Ankara, Turkey
b) Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México, CDMX, 09340 México

Received December 18, 2021, in final form May 16, 2022; Published online May 27, 2022

Abstract
Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space $E_2$ are explored. The study is restricted to Hamiltonians allowing separation of variables $V(x,y)=V_1(x)+V_2(y)$ in Cartesian coordinates. In particular, the Hamiltonian $\mathcal H$ admits a polynomial integral of order $N>2$. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case $N=5$, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case $N>2$ and a formulation of inverse problem in superintegrability are briefly discussed as well.

Key words: integrability in classical mechanics; higher-order superintegrability; separation of variables; exotic potentials.

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