Spinning top in quadratic potential and matrix dressing chain

We show that the equations of motion of the rigid body about centre of mass in the Newtonian field with a quadratic potential are special reductions of period-one closure of the Darboux dressing chain for the Schr\"odinger operators with matrix potentials. We show that the corresponding matrix Schr\"odinger operators are maximally finite-gap (in the sense that for all sufficiently large energies all solutions of the corresponding Schr\"odinger equation are bounded) and describe their spectrum explicitly. The general $2\times 2$-matrix case of the dressing chain, providing also some exotic matrix versions of the harmonic oscillator, is discussed in more detail.