Note on Kowalewski's Top in Quantum Mechanics

It is well known that in classical mechanics algebraic integrals of the top equations only exist in the cases of Euler (asymmetric top, no electric moment), of Lagrange (symmetric top, electric moment parallel to the axis of figure) and of Kowalewski. The latter case is that of a symmetric top whose two equal moments of inertia are twice as large as the third ($A=B=2C$) with an electric moment perpendicular to the axis of figure. The quantum mechanical analogue to Euler's and Lagrange's cases being well known, Kowalewski's case was tried. If $\vartheta$, $\psi$, $\varphi$ are the Euler angles, $Q_{\mathrm{I}}=Q_1+i{Q}_2$, $Q_{\mathrm{II}}=Q_1-i{Q}_2$ linear combinations of the momenta around the principal axes and $U={Q_{\mathrm{I}}}^2+4C\mu sin\vartheta e^{-i\psi },$ then Kowalewski's integral becomes: $U{U}^{*}+U^{*}U+4\mathrm{}{h}^2(Q_{\mathrm{I}}{Q}_{\mathrm{II}}+Q_{\mathrm{II}}{Q}_{\mathrm{I}})=\mathrm{Diag}.\mathrm{Matrix},$ which differs from the classical result by the symmetrization and by the last term proportional to $h^2$.