A class of dynamical symmetries for the Euler-Lagrange equations with the Lagrangian \(L=(1/2)g_{ab}\dot q^ a\dot q^ b\) is determined \((g_{ab}\) are components of Riemannian metric). This class consists of symmetries such that their natural projection onto the configuration manifold yields a vector field or is generated by a totally symmetric tensor field. As it has been recently shown, in general relativity such objects may play the role of generators of the first-integrals of geodesic motion. It is shown in this paper, that the class of tensor fields associated with dynamical symmetries coincides with the family of generalized Killing tensors. Generation of new first integrals through the deformation of a given one via a dynamical symmetry is examined. Correspondent Noether-type conserved quantities are also explicitly exhibited.