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.

A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. In this paper (arXiv:1404.2740), by using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot-Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.