Contact Magnetic Geodesic and Sub-Riemannian Flows on \(V_{n,2}\) and Integrable Cases of a Heavy Rigid Body with a Gyrostat

We prove the integrability of magnetic geodesic flows of \(SO(n)\)-invariant Riemannian metrics on the rank two Stefel variety \(V_{n,2}\) with respect to the magnetic field \(\eta d\alpha\), where \(\alpha\) is the standard contact form on \(V_{n,2}\) and \(\eta\) is a real parameter. Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for \(SO(n)\)-invariant sub-Riemannian structures on \(V_{n,2}\). All statements in the limit \(\eta=0\) imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by \(SO(n)\times SO(2)\)-invariant Riemannian metrics. For \(n=3\), using the isomorphism \(V_{3,2}\cong SO(3)\), the obtained integrable magnetic models reduce to integrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point: the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).