  For $C$ a complex curve and $n \geq 1$, a pair
$(\mathcal{P},\nabla_\mathcal{P})$ of a principal bundle $\mathcal{P}$ with
meromorphic flat connection over $C^n$, holomorphic over the configuration
space $C_n(C)$ of $n$ points over $C$, was introduced in arXiv:1112.0864. For
any point $\infty \in C$, we construct a trivialisation of the restriction of
$\mathcal{P}$ to $(C\setminus\infty)^n$ and obtain a Maurer-Cartan element $J$
over $C_n(C\setminus\infty)$ out of $\nabla_\mathcal{P}$, thus generalising a
construction of Levin and Racinet when the genus of $C$ is higher than one. We
give explicit formulas for $J$ as well as for $\nabla_\mathcal{P}$. When $n=1$,
this construction gives rise to elements of Hain's space of second kind
iterated integrals over $C$.
