Killing tensors on tori
  We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta. 
Konstantin Heil, Andrei Moroianu, Uwe Semmelmann
math.DG