Complex Hamiltonians and integrable systems

The recognition of the heavy top as an invariant subsystem of the elastic problem, which in turn can be seen as a left-invariant optimal control problem on the group of motions of a Euclidean space Eⁿ leads to new insights for a large class of Hamiltonian systems on Lie groups and explains the relevance of the classical theory of tops for problems of optimal control. This paper focuses on the issue of integrability. The main import of the paper is to demonstrate that the classical theory of tops, initiated by L. Lagrange, J. Louiville and S. Kowalewski (1889), extends to holomorphic Hamiltonian systems on complex Lie groups SO_n(C), and that complex Lie groups are a natural setting for proper understanding of the basic phenomena