Magnetic Killing tensors and first integrals of the magnetic flow

In this work we introduce a new family of symmetric tensors generalizing Killing tensors, that we call magnetic Killing tensors. We make use of them to construct first integrals for the magnetic flow associated to a given magnetic field. We apply the results to prove integrability of some invariant magnetic flows (either exact or non-exact) on some 2-step nilmanifolds: the Kodaira-Thurston manifold and Heisenberg nilmanifolds of higher dimensions.