Geometrical classification of Killing tensors on bidimensional flat   manifolds
  Valence two Killing tensors in the Euclidean and Minkowski planes are classified under the action of the group which preserves the type of the corresponding Killing web. The classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebraic. The approach allows to classify both characteristic and non characteristic Killing tensors. 
C. Chanu, L. Degiovanni, R.G. McLenaghan