The master integrable two-dimensional system with a quartic second integral

Abstract

In this paper we construct a new integrable natural conservative mechanical system admitting a second invariant of the fourth degree in velocities. This system is quite general and involves 21 parameters. We also show that all systems with a quartic integral known up to date can be obtained from it as special cases by a relevant choice of the values of parameters. The results are applied to problems of particle and rigid body dynamics. New integrable cases are obtained as special versions of the new system. These cases include motions in a plane, Lobachevsky plane, sphere and surfaces of variable curvature. They also include generalizations of the classical cases of Kovalevskaya, Chaplygin and Goriatchev with the addition of certain singular terms to the potential.