"This project continues and advances the research lines of my past scientific activity, in the context of a reintegration to my home scientific community and the starting of a stable academic activity with a tenure track and a subsequent permanent position.The scientific part of the proposal is the investigation of the geometry of actions of Lie groups in several dynamical and geometrical contexts, with emphasis in their singularities, and is articulated in the following two sections:A. Reduction Theory. Following previous research by me and other groups, I will study the reduction theory for Dirac and generalized complex geometry from both the global and local point of view. This study is a natural continuation of previous research efforts about the reduction theory of symplectic and Poisson manifolds. The main novelty is that we will focus on reduction in the case when the group action presents singularities (fixed points) in Dirac and generalized complex geometry, which is a topic yet to be explored.B. The theory of Hamiltonian relative equilibria. In this section I intend to perform a complete reorganization of the theory of Hamiltonian relative equilibria, as well as to advance it. We will competely redesign the existing theory in a way specifically adapted to the various distinguishing features of symmetric Hamiltonian systems This is a big project started during my previous stage at the University of Manchester. We have substantially advanced this problem during that period, and we expect to have results with significant impact within the duration of this Reintegration Grant."