Necessary and Sufficient Conditions of Kirchhoff Equation Integrability

The problem of motion of a 1-connected solid on interia in an infinite volume of irrotational ideal incompressible liquid in Kirchhoff setting [1-3] is considered in the paper. As it is known, the equations of this problem are structurally analogous to motion equations for the classical problem of motion of a heavy solid around a fixed point. In general case these equations are not integrable as well, and one more additional integral is needed for their integrability. Classical cases of integrability were found by A. Klebsch, V.A. Steklov, A.M. Lyapunov, S.A. Chaplygin in the previous century. It has been shown in [4] that Kirchhoff problems are not integrable in general case, and necessary conditions of integrability, which in some cases are sufficient, have been found there. In the present paper necessary and sufficient conditions of Kirchhoff equations integrability from the view-point of existence of additional analytical and single-valued integrals (in a complex meaning) are investigated.
Analytical results are illustrated with a numerical construction of Poincare mapping and of perturbed asymptotic surfaces (separatrices). Transversal intersection of separatrices may serve as a numerical proof of non-integrability, for great values of pertubing parameter as well.