Holomorphic vector fields with a barycentric condition

We study the $p$-tuples of holomorphic vector fields $(X_1,X_2,\dots ,X_p)$ satisfying the barycentric property ${\displaystyle \sum _k\exp t{X}_k=p\cdot \mathrm{i}\mathrm{d}}$, where $\exp tX$ denotes the flow of $X$.