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 a motion of a heavy rigid body with a gyrostat around a fixed point: 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).