Combinatorics of pre-Lie products sharing the same Lie bracket

We study in detail the operad controlling several pre-Lie products sharing the same Lie bracket. We show that this operad admits a combinatorial description similar to that of Chapoton and Livernet for the pre-Lie operad, namely the underlying species of the operad controlling two pre-Lie products sharing the same Lie bracket is the species of rooted Greg trees, which are rooted trees with black and white vertices such that the black vertices have at least two children, and the composition is the insertion. Moreover, we show that it has many of the remarkable algebraic properties of the pre-Lie operad.