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Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco, Class. Quantum Grav. 25, 165005 (2008)], the Taub-NUT system [A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, and D. Riglioni, SIGMA 7, 048 (2011)], and all the seventeen superintegrable systems for the four types of Darboux spaces as determined in [E.G. Kalnins, J.M. Kress, W. Miller, P. Winternitz, J. Math. Phys. 44, 5811--5848 (2003)].
A procedure to extend a superintegrable system into a new superintegrable one is systematically tested for the known systems on $mathbb E^2$ and $mathbb S^2$ and for a family of systems defined on constant curvature manifolds. The procedure results e
We consider a relativistic charged particle in a background scalar field depending on both space and time. Poincare, dilation and special conformal symmetries of the field generate conserved quantities in the charge motion, and we exploit this to gen
Superintegrable systems on a symplectic manifold conventionally are considered. However, their definition implies a rather restrictive condition 2n=k+m where 2n is a dimension of a symplectic manifold, k is a dimension of a pointwise Lie algebra of a
2nd-order conformal superintegrable systems in $n$ dimensions are Laplace equations on a manifold with an added scalar potential and $2n - 1$ independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenva
Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators but their symmetry algebras d