Abstract: This talk presents new insights into the solutions of the Klein-Gordon equation in the background of a Kerr black hole. First, employing Boyer-Lindquist coordinates for the Kerr metric, it will be derived a constant a motion in addition to the energy and the z-component of the angular momentum of the field which is associated to the Killing tensor of the metric. Second, for the case of vanishing mass of the field, there is given a new symmetry operator S for the solutions with special properties. S is a partial differential operator that commutes with a normal form of the wave operator and contains only partial time derivatives up to the first order. That normal form of the wave operator is obtained by multiplication from the left with the reciprocal of the coefficient function of its second order time derivative. In a canonical formulation of the initial value problem for the wave equation as a PDE system which is first order in the time derivatives, S induces an operator that commutes with the generator of time evolution and therefore very likely generates a one-parameter group of so far unknown symmetries of the solutions of the wave equation.