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Recently a new formalism for perturbations of Maxwell's equations on the background of the Kerr-NUT-(A)dS spacetime was proposed, with which the equations are reduced to a equation of motion of a scalar field that can be solved by separation of variables. In this formalism the differential operators that commute with the operators of the equations of motion, called symmetry operators, played a key role to establish the separable structure. In this work we propose a method to reproduce these commuting symmetry operators in terms of the geometric quantities associated to the hidden symmetry of the background spacetime., Comment: 10 pages. Proceedings for the 11th MSJ-SI "The Role of Metrics in the Theory of Partial Differential Equations"; accepted for publication in Advanced Studies in Pure Mathematics

We show that a spacetime satisfying the linearized vacuum Einstein equations around a type D background is generically of type I, and that the splittings of the Principal Null Directions (PNDs) and of the degenerate eigenvalue of the Weyl tensor are non analytic functions of the perturbation parameter of the metric. This provides a gauge invariant characterization of the effect of the perturbation on the underlying geometry, without appealing to differential curvature invariants. This is of particular interest for the Schwarzschild solution, for which there are no signatures of the even perturbations on the algebraic curvature invariants. We also show that, unlike the general case, the unstable even modes of the Schwarzschild naked singularity deforms the Weyl tensor into a type II one., Comment: 9 pages