Integrability of the Kirchhoff equations in the symmetrical case (Q1073271)

The integrability of the Kirchhoff equations is investigated in the symmetric case when these equations are equivalent to the Euler-Poisson equations of the movement of the solid with motionless point in the axial-symmetric force field U(\(\gamma)\). The Hamiltonian is \[ {\mathcal H}=(1/2)<I\omega,\omega >-U(\gamma),\quad U=(1/2)\sum^{3}_{i,j=1}c_{ij}\gamma_ i\gamma_ j, \] where \(\omega\) is the angular velocity vector, \(I=diag(a_ 1^{-1},a_ 2^{-1},a_ 3^{-1})\), \(\gamma\) is the unit vector along the symmetry axis of the force field. Using the Poincaré method of asymptotic surfaces splitting it is shown that if \(a_ 1=a_ 2\neq a_ 3\) and the symmetric matrix C has elements \(c_{11}=c_{22}+\epsilon\), \(c_{22}\neq c_{33}\), \(\epsilon c_{12},\epsilon c_{13},\epsilon c_{23}\), the Kirchhoff equations (being completely integrable in the case \(\epsilon =0)\) for small \(\epsilon\) \(\neq 0\) do not possess the fourth integral, analytical in \(R^ 3\{\omega \}\times R^ 3\{\gamma \}\).