CYK tensors, Maxwell field and conserved quantities for the spin-2 field

Starting from an important application of Conformal Yano--Killing tensors for the existence of global charges in gravity, some new observations at \⁺ are given. They allow to define asymptotic charges (at future null infinity) in terms of the Weyl tensor together with their fluxes through \⁺. It occurs that some of them play a role of obstructions for the existence of angular momentum. Moreover, new relations between solutions of the Maxwell equations and the spin-2 field are given. They are used in the construction of new conserved quantities which are quadratic in terms of the Weyl tensor. The obtained formulae are similar to the functionals obtained from the Bel--Robinson tensor.