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Symmetry analysis of the charged squashed Kaluza-Klein black hole metric

In this paper, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza--Klein black hole spacetime in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries are presented and then it the $n$-dimensional optimal system of this space-time metric, for $n=1,\ldots,4$, are computed. It is shown that there is not any $n$-dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for $n\geq5$. Then the point symmetries of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem.