Completion of vector fields and the Painlevé property (Q1188226)

The arena of this important study is the category of complex analytic dynamical systems with (infinitesimally) equivariant holomorphic mappings as morphisms. The author develops a systematic method for detecting ``completability'' of arbitrary flows. His main results include: (1) On a symplectic manifold with a strictly completable Hamiltonian field, the symplectic structure as well as the Hamiltonian and all its integrals extend to the completion. (2) A ``Zyglin-analysis'' type necessary condition for completability involving the monodromy of the variational equations. (3) A ``Painlevé analysis'' type necessary condition for strict completability also involving monodromy and its application to the Henon- Heiles' two parameter family of Hamiltonians. (4) A list of canonical forms for all complete fields on an arbitrary Riemann surface. (5) Every 1-dimensional system has a (not necessarily strict) completion. (6) A detailed description of the situation for meromorphic fields \(\mathbb{P}^ 1\).