In this paper (arXiv:1311.0792) we study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We provide the complete local classification of Lie-Hamilton systems on the plane and we study new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

In this paper (arXiv:1310.6554) a new maximally superintegrable deformation of the N-dimensional Kepler–Coulomb Hamiltonian is presented. From a geometric viewpoint, this superintegrable Hamiltonian can be interpreted as a system on an N-dimensional Riemannian space with nonconstant curvature. The eigenvalues and eigenfunctions of the model are explicitly obtained, and the spectrum presents a hydrogen-like shape for positive values of the deformation parameter and of the corresponding coupling constant.

The Proceedings of the XXI IFWGP, that our group organized in Burgos last year, have been already published as the September 2013 issue of the International Journal of Geometric Methods in Modern Physics.

A Lie system is a nonautonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie-Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. In this new paper (arXiv:1305.6272) is shown that Lie-Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods to derive in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer-Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky-Winternitz systems with time-dependent frequency.

We present an overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces (arXiv:1304.4544). It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.