Completely integrable symplectic mapping

1o Introduction. The complete integrability, whose concept was first introduced by Liouville, is still an important subject in the recent develop- ment of Hamiltonian systems 1, This. concept exists not only in Hamiltonian mechanical systems but also in certain systems of evolution equations. Also, the complete integrability can be found in symplectic mappings, which is the theme treated in this paper. Though any ex- ponential map of a Hamiltonian vector field isa symplectic mapping, the converse is not true. Therefore, it is not always the case that a symplectic mapping possesses properties similar to those of a Hamiltonian flow. As