Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras

We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere $S^{3}$ using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic and higher order polynomial algebras in context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.Comment: 15 page

Integrable and superintegrable systems with spin

A system of two particles with spin s=0 and s=1/2 respectively, moving in a plane is considered. It is shown that such a system with a nontrivial spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order integrals of motion. The Pauli equation is solved in this superintegrable case and reduced to a system of ordinary differential equations when only one first-order integral exists.Comment: 12 page

Exact Solvability of Superintegrable Systems

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the $sl(3)$-algebra, realized by first order differential operators.Comment: 14 pages, AMS LaTe

Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion

We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in cartesian coordinates with a third order integral are known. The general formalism is applied to one of the quantum potentials

Third-order superintegrable systems separable in parabolic coordinates

In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equations and show that all such systems are multi-separable and so admit two second-order integrals. The third-order integral is their Lie or Poisson commutator. We discuss how this situation is different from the Cartesian and polar cases where new potentials were discovered which are not multi-separable and which are expressed in terms of Painlev\'e transcendents or elliptic functions

Lie Point Symmetries and Commuting Flows for Equations on Lattices

Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice

Generalized Kadomtsev-Petviashvili equation with an infinite dimensional symmetry algebra

A generalized Kadomtsev-Petviashvili equation, describing water waves in oceans of varying depth, density and vorticity is discussed. A priori, it involves 9 arbitrary functions of one, or two variables. The conditions are determined under which the equation allows an infinite dimensional symmetry algebra. This algebra can involve up to three arbitrary functions of time. It depends on precisely three such functions if and only if it is completely integrable.Comment: AMSLaTeX, 16 pages, no figures, corrected some typos and added two new section