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We give an overview of the first integrals of motion of particles in thepresence of external gauge fields in a covariant Hamiltonian approach. Thespecial role of St\"ackel-Killing and Killing-Yano tensors is pointed out. Somenontrivial examples involving Runge-Lenz type conserved quantities areexplicitly worked out. A condition of the electromagnetic field to maintain thehidden symmetry of the system is stated. A concrete realization of thiscondition is given by the Killing-Maxwell system and exemplified with the Kerrmetric. Quantum symmetry operators for the Klein-Gordon and Dirac equations areconstructed from Killing tensors. The transfer of the classical conservedquantities to the quantum mechanical level is analyzed in connection withquantum anomalies.

Covariant Approach of the Dynamics of Particles in External Gauge Fields, Killing Tensors and Quantum Gravitational Anomalies