of the Kontsevich star-product formula. This is in some sense the case. Namely we can restate the integrability problem into a formal integration problem. The solution of this problem is called the **formal symplectic groupoid** over a Poisson manifold which is a formal version of the “true symplectic groupoid" that exists however even for non-integrable Poisson structures. This is exactly what the question marks stand for in the above table. Let us be more precise.

Formal integration problem for Poisson manifolds
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In the sequel we will only consider Poisson structures $\alpha$ over $M=\mathbb R^d$. Suppose that $(M,\alpha)$ is integrable and that its symplectic groupoid $G$ satisfies the two following properties (which are always satisfied in a neighborhood of $M$):

1.  $ G\subset T^*M\simeq \mathbb R^{*d} \times \mathbb R^d$

2.  $G^{(m)}\subset \overline{T^*M}\times \overline{T^*M}\times  T^*M$ is an exact Lagrangian manifold, i.e., there exists a generating function $S:\mathbb R^{*d}\times \mathbb R^{*d}\times \mathbb R^d\rightarrow \mathbb R$ such that $G^{(m)}= {\operatorname{graph}}(dS)$.

We would like to see what sort of constraints the associativity of the groupoid product imposes on $S$. First of all we may remark that under the previous assumptions the product space $G^{(m)}$ can be described as follows $$G^{(m)} = \Big\{\Big(
\big(  p_1,\nabla_{p_1}S\big),
