An invariant classification of cubic integrals of motion

We employ an isometry group invariants approach to study Killing tensors of valence three defined in the Euclidean plane. The corresponding invariants are found to be homogeneous polynomials of the parameters of the vector space of the Killing tensors. The invariants are used to classify the non-trivial first integrals of motion which are cubic in the momenta of Hamiltonian systems defined in the Euclidean plane. The integrable cases isolated by Holt and Fokas-Lagerstrom are investigated from this viewpoint.