In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.

We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid.

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We study the geometry of completely integrable bi-Hamiltonian systems, and in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.KEY WORDS: Bi-Hamiltonian system; completely integrable system. IntroductionThe study of completely integrable Hamiltonian systems, i.e., systems admitting a complete sequence of first integrals, started with the pionneering work of Liouville (1855) on finding local solutions by quadratures. We have now a complete picture of the geometry of such systems, which in its modern presentation is due to Arnol'd (1988). A major flaw in the Arnol'd-Liouville theory is that it provides no indication on how to obtain first integrals, and this is one of the main reasons for the growing interest on bi-Hamiltonian systems (systems admitting two compatible Hamiltonian formulations). For a given bi-Hamiltonian system, a result due to Magri (1978) shows how to construct a whole hierarchy of first integrals. Under an additional assumption, one can show that Magri's theorem yields a complete sequence of first integrals. Moreover, this assumption may be formulated in a way that still makes sense in the setting of infinite dimensional systems. Therefore, if one wants to extend the notion of complete integrability, the following natural question arises: Given a completely integrable Hamiltonian system, does the complete sequence of first integrals arise from a second Hamiltonian structure via Magri's theorem? This problem was first studied by Magri and Morosi (1984) in a set of unpublished notes, which seems to contain an incorrect answer. More recently, Brouzet in his "thése de doctorat" studied the same question when the dimension equals four, and showed that the answer in general is negative (Brouzet, 1990). Our work might be considered an improvement on their results. Theorem 3.1 below shows that a second Hamiltonian structure exists if and only if the Hamiltonian function satisfies a certain geometric condition. This