A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.

On superintegrable systems with a position-dependent mass in polar-like coordinates

For a superintegrable system defined in plane polar-like coordinates introduced by Szumiński et al. and studied by Fordy, we show that the system with a position-dependent mass is separable in three distinct coordinate systems. The corresponding separation equations and additional integrals of motion are derived explicitly. The closure algebra of integrals is deduced. We also make a generalization of this system by employing the classical Jacobi method. Lastly a sufficient condition which ensures flatness of the underlying space is derived via explicit calculation.

Deformed oscillator algebra for quantum superintegrable systems in two-dimensional Euclidean space and on a complex two-sphere

In this work, we study superintegrable quantum systems in two-dimensional Euclidean space and on a complex twosphere with second-order constants of motion. We show that these constants of motion satisfy the deformed oscillator algebra. Then, we easily calculate the energy eigenvalues in an algebraic way by solving of a system of two equations satisfied by its structure function. The results are in agreement to the ones obtained from the solution of the relevant Schroedinger equation.

Nonlinear dynamical symmetries of Smorodinsky-Winternitz and and Fokas-Lagerstorm systems

General solutions of the Smorodinsky-Winternitz system and the Fokas-Lagerstorm system, which are superintegrable in two-dimensional Euclidean space, are obtained using the algebraic method （structure function）. Their dynamical symmetries, which are governed by deformed angular momentum algebras, are revealed.

Degeneration of the Superintegrable System with Potentials Described by the Sixth PainlevéTranscendents

This article concerns the quantum superintegrable system obtained by Tremblay and Winternitz, which allows the separation of variables in polar coordinates and possesses three conserved quantities with the potential described by the sixth Painlevé equation. The degeneration procedure from the sixth Painlvé equation to the fifth one yields another new superintegrable system;however, the Hermitian nature is broken.

NEW SUPER-HAMILTONIAN STRUCTURES OF SUPER-DIRAC HIERARCHY

In this paper, we introduce the supertrace identity and its applications. A new eight-dimensional Lie superalgebra is constructed and the super-Dirac hierarchy is derived. By the supertrace identity, we obtain the super-bi-Hamiltonian structure of the super-Dirac hierarchy.