Integrability conditions for Kirchhoff equations (Q1813876)

Kirchhoff's equations describe the motion of a rigid body by inertia in an ideal incompressible fluid. Let \(M\) be the complete momentum of the system ``body + fluid'', and \(p\) be the momentum of the system. The author considers quadratic Hamiltonians \(H=\tfrac 12 \langle aM,M\rangle +\langle bM,p\rangle+\tfrac 12 \langle cp,p\rangle\) with diagonal matrices \(a\), \(b\), \(c\) and the symmetry \(a_1=a_2\). For such systems necessary conditions are found for the existence of the additional meromorphic integral. An integrability criterion for the body with three symmetry planes is described in the paper.