We present (arXiv:1605.02384) a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the two-dimensional sphere and to the hyperbolic space of the two-dimensional anisotropic oscillator with any pair of frequencies. The new curved Hamiltonian depends on the curvature κ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of the system relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superintegrable for commensurate frequencies, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the κ→0 limit. The additional constant of motion in the commensurate case is, as expected, of higher-order in the momenta and can be explicitly deduced by performing the classical factorization of the Hamiltonian. The known 1:1 and 2:1 anisotropic curved oscillators are recovered as particular cases, meanwhile all the remaining ωx:ωy curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the quantum Hamiltonian turns out to be quantum superintegrable and leads to a new exactly solvable quantum model. Its corresponding spectrum, that exhibits a maximal degeneracy, is explicitly given as an analytical deformation of the Euclidean eigenvalues in terms of both the curvature κ and the Planck constant ℏ. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.

In the recent paper arXiv:1601.03357, a method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson-Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rössler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson-Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.

In this paper (arXiv:1512.06610) the factorization technique to superintegrable systems is revisited. We recall that if an integrable classical Hamiltonian H can be separated in a certain coordinate system, it is well known that each coordinate leads to an integral of the motion. Then, for each coordinate two sets of “ladder” and “shift” functions can be found. It is shown that, if certain conditions are fulfilled, additional constants of motion can be explicitly constructed in a straightforward manner by combining these functions, and such integrals are, in the general case, of higher-order on the momenta. We apply this technique to both known and new classical integrable systems, and we stress that the very same procedure can also be applied to quantum Hamiltonians leading to ladder and shift operators. In particular, we study the factorization of the classical anisotropic oscillators on the Euclidean plane and by making use of this technique we construct new classical (super)integrable anisotropic oscillators on the sphere. Finally, we also illustrate this approach through the well-known Tremblay-Turbiner-Winternitz (TTW) system on the Euclidean plane.

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. In this paper (arXiv:1511.08908) we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.