  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.
