On the Integrable Deformations of the Maximally Superintegrable Systems

In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new maximally superintegrable systems are obtained. We also consider the particular case of Hamiltonian mechanical systems. In addition, we use this method to construct some deformations of an arbitrary system of first-order autonomous differential equations.

Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

A non-degenerate second-order maximally conformally superintegrable system in dimension 2 naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's Stäckel class can be obtained from this associated quadric.The Stäckel class of a second-order maximally conformally superintegrable system is its equivalence class under Stäckel transformations, i.e., under coupling-constant metamorphosis.

Carter's constant and superintegrability

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.

Dirac equation with anisotropic oscillator, quantum E3′ and Holt superintegrable potentials and relativistic generalized Yang–Coulomb monopole system

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum [Formula: see text], anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a [Formula: see text] non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.