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.

In this paper (arXiv:1411.7569) we review two maximally superintegrable Hamiltonian systems that are defined, respectively, on an N-dimensional spherically symmetric generalization of the Darboux surface of type III and on an N-dimensional Taub-NUT space. Afterwards, we show that their quantization leads, respectively, to exactly solvable deformations of the two basic quantum mechanical systems: the harmonic oscillator and the Coulomb problem. In both cases the quantization is performed in such a way that the maximal superintegrability of the classical  Hamiltonian is fully preserved.  In particular, we prove that this strong condition is fulfilled by  applying the so-called conformal Laplace-Beltrami quantization prescription. In this way, the eigenvalue problems for the quantum counterparts of these two Hamiltonians can be rigorously solved, and it is found that their discrete spectrum is just a smooth deformation of the oscillator and Coulomb spectrum, respectively. Moreover, it turns out that the maximal degeneracy of both systems is preserved under the deformation induced by the curvature. Finally, new multiparametric generalizations of both systems that preserve their superintegrability are envisaged.

In this paper (arXiv:1411.2033) we construct a constant curvature analogue  on the two-dimensional sphere and  the hyperbolic space of the integrable Hénon-Heiles Hamiltonian of KdV type. The curved integrable Hamiltonian so obtained depends on a real parameter which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system is smoothly obtained in the zero-curvature limit. On the other hand, the Hamiltonian that we propose can be regarded as an integrable perturbation of a known curved integrable 1:2 anisotropic oscillator. We stress that in order to obtain the curved  Hénon-Heiles Hamiltonian,  the preservation of the full integrability structure of the flat Hamiltonian under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani series of integrable polynomial potentials, in which the flat Hénon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved Ramani potentials will be also explicitly presented.

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. In this new paper (arXiv:1410.7336), after reviewing the classification of finite-dimensional real Lie algebras of Hamiltonian vector fields on the plane, we present new Lie-Hamilton systems with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the hereafter called planar diffusion Riccati systems and complex Bernoulli equations, all of them with t-dependent real coefficients. Furthermore, we study the existence of local diffeomorphisms among new and already known Lie-Hamilton systems on the plane. In particular, we show that the Cayley-Klein Riccati equations describe as particular cases well-known coupled Riccati equations, second-order Kummer-Schwarz equations, Milne-Pinney equations, the harmonic oscillator with t-dependent frequency and other systems of physical and mathematical relevance.

In this new paper (arXiv:1408.3689) it is shown that the canonical classical r-matrix arising from the Drinfel’d double structure underlying the two-fold centrally extended (2+1) Galilean and Newton-Hooke Lie algebras (with either zero or non-zero cosmological constant Λ, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical r-matrices associated with the Drinfel’d double structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and Newton-Hooke Drinfel’d doubles is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting ‘absolute time’ coordinate together with two noncommutative space coordinates, whose commutator is a function of the cosmological constant Λ and of the (central) ‘quantum time’ coordinate. Thus, the Chern-Simons approach to Galilean (2+1) gravity can be consistently understood as the appropriate non-relativistic limit of the Lorentzian theory, and their associated quantum group symmetries (which do not fall into the family of so-called kappa-deformations) can also be derived from the (anti)-de Sitter quantum doubles through a well-defined quantum group contraction procedure.

In this paper (arXiv:1407.1401) we quantize an N-dimensional classical Hamiltonian system that can be regarded as a deformation of the Coulomb problem. Moreover, the kinetic energy term in the Hamiltonian is just the one corresponding to an N-dimensional Taub-NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the classical Hamiltonian is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose as the kinetic part of the Hamiltonian the conformal Laplacian of the underlying Riemannian manifold, which combines  the usual Laplace-Beltrami operator on the Taub-NUT manifold and a multiple of its scalar curvature. As a consequence, we obtain a novel exactly solvable deformation of the quantum Coulomb problem, whose spectrum is computed in closed form, and showing that the well-known maximal degeneracy of the flat system is preserved in the deformed case. Several interesting algebraic and physical features of this new exactly solvable quantum system are analysed.