Symmetries and Exact Solutions in General Relativity

In this text, symmetries are studied in the context of general relativity. Initially, we address the problem of the separability of the Hamilton-Jacobi equation for the geodesic Hamiltonian. In this scenario, the existence of classes of coordinate systems that separate the Hamilton-Jacobi equation is intimately connected to the existence of symmetries. In fact, the close relationship between the notion of separability structures and the existence of symmetries is manifest in this framework. In particular, we show that the existence of a separability structure enables the complete integrability of the geodesic motion. Then, a study on symmetries from the point of view of the action of continuous group on differential manifolds is conducted. To this end, we review groups, Lie groups and Lie algebras. Finally, equipped with all this knowledge, starting with the most general four-dimensional spacetime possessing two commuting Killing vectors and a nontrivial Killing tensor, we analytically integrate Einstein-Yang-Mills equations for a completely arbitrary gauge group. In particular, generalizations of the Kerr-NUT-(A)dS spacetime containing nonabelian gauge fields are obtained.