The search of hidden symmetries, photon surfaces, and black hole shadows

Hidden symmetries supply dynamical systems with integrals of motion that can be represented as polynomials in conjugate momenta. Conformal Killing tensors (CKT) make it possible to obtain such integrals of motion for isotropic geodesics and a massless scalar field on the background of a particular spacetime. It turns out to be extremely useful to project the CKTs onto the space-time slices. The tensor projected onto a slice often has a trivial form that can be used to search for and generate unknown CKTs in the entire spacetime. The slices of space-time bundle that allow generating CKTs turn out to be closely related to the photonic structure of space-time, which generates a relationship between the generated CKTs and the shadows of black holes and other ultra-compact objects in the theory of gravity. I will consider a theorem that allows generating CKTs by solving simple algebraic equations and linear differential equations of the first order, their relation to photonic surfaces and shadows of black holes, as well as the case of exact Killing tensors.