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.

In this work (arXiv:1503.09187) the constant curvature analogue on the two-dimensional sphere and the hyperbolic space of an integrable Hénon-Heiles Hamiltonian of KdV type, is revisited. The resulting integrable curved Hamiltonian depends on a parameter κ which is just the curvature of the underlying space and allows one to recover the initial Hamiltonian under the smooth flat/Euclidean limit κ→0. This system can be regarded as an integrable cubic perturbation of a specific curved 1:2 anisotropic oscillator, which was already known in the literature. The Ramani series of potentials associated to the curved Hamiltonian is fully constructed, and corresponds to the curved integrable analogues of homogeneous polynomial perturbations of H that are separable in parabolic coordinates. Integrable perturbations are also presented, and they can be regarded as the curved counterpart of integrable rational perturbations of the Euclidean Hénon-Heiles Hamiltonian. It is explicitly shown that the latter perturbations can be understood as the “negative index” counterpart of the curved Ramani series of potentials. Furthermore, it is shown that the integrability of the curved Hénon-Heiles Hamiltonian is preserved under the simultaneous addition of curved analogues of “positive” and “negative” families of Ramani potentials.

In this work (arXiv:1502.07518) we present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern–Simons formulation of (2+1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel’d double structures, we believe said structures are worth being analysed also in the (3+1) scenario as a possible guiding
principle towards the description of (3+1) gravity. To this aim, a canonical classical r-matrix arising from a Drinfel’d double structure for the three (3+1) Lorentzian algebras is obtained. This r-matrix turns out to be a twisted version of the one corresponding to the (3+1) kappa-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the kappa-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.

A Lie system is a system of differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. In this paper (arXiv:11412.0300) we define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on R and R^2. Our results shall be illustrated through examples of physical and mathematical interest.