Degenerate metrics on a dual geometry of spherically symmetric space-time

We consider a massive spinless particle in a particular curved space-time described by a phase space formalism. For this space-time, we determine nontrivial conserved quantities and geometrical invariants by analytical and numerical methods. By analytic calculations, we revisit (perform) a method to relate two kinds of geometries associated with conserved quantities, whose dual metrics are constructed. The method relies on the calculation of the Stackel-Killing tensors (SKT) presented in a spherically symmetric space-time. We have a system of partial differential equations (PDE) for the symmetric components of the SKT obtained for a given space-time metric. We note that the PDE system is highly non-trivial according to the isometry structure of the original metric. A degenerate metric solution appears in the dual structure, and it is compared with recent works involving space-time bridge solutions in vacuum gravity and nontrivial extensions of the Schwarzschild space-time.