Gauge symmetries and isometries in higher dimensions

We study the invariance properties of five-dimensional metrics and their corresponding geodesic equations of motion. In this context a number of five-dimensional models of the Einstein–Gauss–Bonnet (EGB) theory leading to black holes, wormholes and spacetime horns arising in a variety of situations are discussed in the context of variational symmetries of which each vector field, via Noether’s theorem (NT), provides a nontrivial conservation law. In particular, it is shown that algebraic structure of isometries and the variational conservation laws of the five-dimensional Einstein–Bonnet metric extend consistently from the well-known Minkowski, de-Sitter and Schwarzschild four-dimensional spacetimes to the considered five-dimensional ones. In the equivalent five-dimensional case, the maximal algebra of kvs is fifteen with eight additional Noether symmetries. Also, whereas the constant curvature five-dimensional case leads to fifteen kvs and one additional Noether symmetry and seven plus one in the minimal case, a number of metrics of the EGB theory in five dimensions give rise to algebras isomorphic a seven-dimensional algebra of kvs and a single additional Noether symmetry.

Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.