I am trying to find three integrable systems with 6 degrees of freedom using the Liouville–Arnold theorem. That means that a set of integrals of motion that correspond to a conserved quantity for every degree of freedom needs to be found. Additionally, these integrals of motion need to have vanishing Poisson brackets with each other. I was able to prove that a 6-dimensional harmonic oscillator works, as well as the motion of a two-body system that's sitting in a central potential. For the latter system, one can find the conserved quantities that constitude such a set in relative and centre-of-motion coordinates: The energy, the three components of the momentum, the square of the angular momentum and one of the components of the angular momentum.
Now, similarly to the two-body problem, I thought of a rigid body in 3D-space that's able to move and rotate freely in all directions. I feel like the conserved quantities should be the exact same. However, I am having trouble finding proof for that claim. I don't know how I would transform the Hamiltonian of the system so that any conserved quantities would become obvious.
