  In this work, we characterize all the point processes $\theta=\sum_{i\in
\mathbb{N}} \delta_{x_i}$ on $\mathbb{R}$ which are left invariant under
branching Brownian motions with critical drift $-\sqrt{2}$. Our
characterization holds under the only assumption that
$\theta(\mathbb{R}_+)<\infty$ almost surely.
