A unified treatment of cubic invariants at fixed and arbitrary energy

Cubic invariants for two-dimensional Hamiltonian systems are investigated u sing the Jacobi geometrization procedure. This approach allows for a unifie d treatment of invariants at both fixed and arbitrary energy. In the geomet ric picture the invariant generally corresponds to a third rank Killing ten sor, whose existence at a fixed energy value forces the metric to satisfy a nonlinear integrability condition expressed in terms of a Kahler potential . Further conditions, leading to a system of equations which is overdetermi ned except for singular cases, are added when the energy is arbitrary. As s olutions to these equations we obtain several new superintegrable cases in addition to the previously known cases. We also discover a superintegrable case where the cubic invariant is of a new type which can be represented by an energy-dependent linear invariant. A complete list of all known systems which admit a cubic invariant at arbitrary energy is given. (C) 2000 Ameri can Institute of Physics. [S0022-2488(00)01101-4].