  Two-dimensional Hamiltonian systems admitting second invariants which are
quartic in the momenta are investigated using the Jacobi geometrization of the
dynamics. This approach allows for a unified treatment of invariants at both
arbitrary and fixed energy. In the differential geometric picture, the quartic
invariant corresponds to the existence of a fourth rank Killing tensor.
Expressing the Jacobi metric in terms of a Kahler potential, the integrability
condition for the existence of the Killing tensor at fixed energy is a
non-linear equation involving the Kahler potential. At arbitrary energy,
further conditions must be imposed which lead to an overdetermined system with
isolated solutions. We obtain several new integrable and superintegrable
systems in addition to all previously known examples.
