homogeneous part of $S_n$ of degree $i$ in the first argument.

In Section \[recovering\] we show that, provided we have a natural deformation $S_h=S_0 +hS_1+h^2S_2+\dots$ of the trivial generating function, we can deform the structure maps of the trivial symplectic groupoid into $$\begin{array}{cccc}
\epsilon_h(x) & = & (0,x)  &\textbf{unit map}\\
i_h(p,x)      & = & (-p,x) &\textbf{inverse map}\\
s_h(p,x)      & = & \nabla_{p_2}S_h(p,0,x) & \textbf{source map}\\
t_h(p,x)      & = & \nabla_{p_1}S_h(0,p,x) & \textbf{target map}
\end{array}$$ such that the groupoid structure is (formally) preserved.

Moreover there is a unique Poisson bracket on $\mathbb R^d$ such that the source, $s_h$, is a Poisson map with respect to the canonical symplectic structure on the formal symplectic groupoid. This Poisson bracket is given by $\{f,g\}_{\mathbb R^d}(x)=2hS_1(df,dg,x)$, the first order term of the generating function. We can now formulate the formal integration problem for Poisson manifolds.

**Formal integration problem for Poisson manifolds:**

*Given a Poisson structure on $\mathbb R^d$, does there exists a deformation of the trivial generating function such that the first order term is the original Poisson structure?*

Main Result, main example, main interpretation
----------------------------------------------

The following theorem gives a positive answer to the deformation problem for symplectic groupoids. This is the main result 