A manifold structure for analytic isotropy Lie pseudogroups of infinite type

We give the solution of Lie's third fundamental problem for the class of infinite dimensional Lie algebras corresponding to the isotropy sub-pseudogroups of the flat transitive analytic Lie! pseudogroups of infinite type. The associated Lie groups are regular Gateaux-analytic infinite-dimensional Lie groups whose compatible manifold structure is modelled on locally convex topological vector spaces (countable inductive limits of Banach spaces) of vector fields by charts involving countable products exponential mappings. This structure theorem is applied to the local automorphisms pseudogroups of Poisson, symplectic, contact and unimodular structures. In particular the local analytic Lie-Poisson algebra associated to any finite dimensional real Lie algebra is shown to be integrable into a unique connected and simply connected regular infinite-dimensional Gateaux-analytic Lie group.