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).