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We introduce a systematic method to solve a type of Cartan’s realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We illustrate our method with the problem of classification of extremal Kähler metrics on surfaces. 

We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it. 

Abstract This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompassesseveral classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamentalproperties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like thede Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.)and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTsare related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivialPMCT related to a classical question on the topology of the orbits of a free symplectic circle action.In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry,Hamiltonian G -spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.