The solution of this formal integration problem relies on the existence of a generating function for which we give here the explicit formula. This generating function turns out to be a universal Campbell–Baker–Hausdorff(CBH) formula for the non linear case. It reduces to the usual CBH formula when the Poisson structure comes from a Lie algebra. This generating function can be interpreted as the semi-classical part of Kontsevich deformation quantization formula. This fact reminds the origin of symplectic groupoids which were first introduced by Weinstein in [@CDW1987], Karasev in [@karasev1987], and Zakrwewski in [@zakrzewski1990] as a tool to quantize the algebra of functions on a Poisson manifold. This section is devoted to recall some basic features of the program of quantization by symplectic groupoid, to formulate the formal integration problem for Poisson manifolds and to state the main theorem of this article which gives a positive answer to the formal integration problem.

Quantization by symplectic groupoid
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The program of quantization by symplectic groupoid is an attempt to quantize the algebra of functions on a Poisson manifolds by geometric means.

It is based mainly on the belief or hope, coming from geometric quantization, that there should exist a kind of correspondence 