Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body

The solutions to the Euler-Poisson equations are geodesic lines of $SO(3)$ manifold with the metric determined by the inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra $e(3)$. The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler-Poisson equations is written in terms of exponential of the Hamiltonian vector field.