New Integrable Systems with a Quartic Integral and New Generalizations of Kovalevskaya’s and Goriatchev’s Cases

In his paper [1], one of us has introduced a method for constructing integrable conservative two-dimensional mechanical systems, on Riemannian 2D spaces, whose second integral is a polynomial in the velocities. This method was applied successfully in [2] to construction of systems admitting a cubic integral and in [3, 4] and [5] to cases of a quartic integral. The present work is devoted to construction of new integrable systems with a quartic integral. The potential is assumed to have the structure $V = u(y) + v(y) (a \cos x + b \sin x) + w(y) (c \cos 2x + d \sin 2x)$. This is inspired by the structure of potential in the famous generalization of Kovalevskaya’s case in rigid body dynamics introduced by Goriatchev. The resulting differential equations were completely solved only for time reversible systems. A 10-parameter family of systems of the searched type is obtained. Four parameters determine the structure of the line element of the configuration manifold and the others contribute only to the potential function. In the case of time-irreversible systems the governing equations were solved in the three cases when the metric is identical to that of reduced rigid body motion. Those lead to three new several-parameter generalizations of known cases, including the classical cases of Kovalevskaya, Chaplygin and Goriatchev.