Two-dimensional conservative mechanical systems with quartic second integral

The method introduced in [20] was applied in [21] and [22] for constructing integrable conservative two dimentional mechanical systems whose second integral of motion is polynomial up to third degree in the velocities. In this paper we apply the same method for systemaic construction of mechanical systems with a quartic integral. As in our previous works, the configuration space is not assumed an Euclidean plane. This widens the range of applicability of the results to diverse mechanical systems such as rigid body dynamics and motion on two dimensional surfaces of positive, negative and variable curvature. Two new several-parameter integrable systems are obtained, which unify and generalize several previously known ones by modifying the configuration manifold and the potential of the forces acting on the system. Those systems are shown to include as special cases, integrable problems of motion in the Euclidean plane, the hyperbolic plane and different types of curved two dimensional manifolds. The results are applied to problems of rigid body dynamics. New integrable cases are obtained as special versions of one of the new systems, corresponding to different choices of the parameters. Those cases include new generalizations of the classical cases of Kovalevskaya, Chaplygin and Goryachev.