New Families of Integrable Two-Dimensional Systems with Quartic Second Integrals

The method introduced in [11] and [12] is extended to construct new families of severalparameter integrable systems, which admit a complementary integral quartic in the velocities. A list of 14 systems is obtained, of which 12 are new. Each of the new systems involves a number of parameters ranging from 7 up to 16 parameters entering into its structures. A detailed preliminary analysis of certain special cases of one of the new systems is performed, aimed at obtaining some global results. We point out twelve combinations of conditions on the parameters which characterize integrable dynamics on Riemannian manifolds as configuration spaces. Very special 7 versions of the 12 cases are interpreted as new integrable motions with a quartic integral in the Poincaré half-plane. A byproduct of the process of solution is the construction of 12 Riemannian metrics whose geodesic flow is integrable with a quartic second integral.