Heavy rigid body with a gyroscope in $\mathbb R^n$

Starting from the following multidimensional integrable generalizations of the heavy rigid body systems: the Euler top, the Lagrange top, the Lagrange bitop, and the totally symmetric case, we add to each of them a gyroscope. For each of the newly constructed systems, we provide a polynomial matrix Lax representation and prove Liouville integrability.