Kowalewski basis for the hydrogen atom

The completely integrable problem of Kowalewski's top in classical mechanics is extended to the groups $O(4)$ and $O(3,1)$. For each classical system on the groups $O(4)$, $E(3)$, $O(3,1)$ three inequivalent quantum analogs are found. For the Coulomb problem, this results in the construction of one Kowalewski basis in classical mechanics and three in quantum mechanics.