We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by $\circledast$ the branching convolution operation introduced by Bertoin and Mallein (2019), and by $\mathcal{Z}$ the law of a random point measure on the real line, we are interested in solutions to the fixed point equation \[ \mathcal E = \mathcal{Z} \circledast \mathcal E, \] with $\mathcal E$ a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.