Our research is aimed to apply different **algebraic and geometric structures **such as Lie algebras and groups, representation theory, homogeneous spaces, spacetime symmetries, Hopf algebras, quantum groups, Lie bialgebras, Poisson-Lie groups, Poisson algebras and Poisson geometry, integrability theory, quantization techniques and noncommutative geometry, for the study of ** classical and quantum Hamiltonian systems, **with special emphasis on their** symmetry and integrability properties**.

Nowadays, our principal research subjects are the following:

- Classical and quantum (super)integrable Hamiltonian systems on curved spaces.
- Bertrand´s theorem for spaces with nonconstant curvature.
- Hamiltonian structure and integrability properties of Lotka-Volterra systems.
- Integrable Hamiltonian systems with quantum group symmetry.
- Geometry and representation theory of the symmetry groups of spacetimes.
- Contractions of Lie algebras, groups and quantum groups.
- Classification of quantum deformations of spacetime symmetries.
- Lie bialgebra quantization techniques.
- Non-commutative spacetimes with cosmological constant.
- Quantum groups and Poisson-Lie groups in (2+1) gravity.