We consider a particle system on the real line in which each particle evolves into many particles via independent Branching Brownian motions. Under a very mild natural assumption, we give a full characterisation of the fixed points of this particle system. This result is motivated by an earlier work of T. Liggett on particle systems without branching. Further motivations come from the close connection between the Branching Brownian motion and the two dimensional Gaussian free field. This talk is based on a joint work with Christophe Garban and Xinxin Chen.