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Abstract: In this talk, we will present some results concerning the behavior of solutions to the one-phase Bernoulli problem that are modeled —either at infinitesimally or at infinity—by one-homogeneous solutions with isolated singularities. We address the uniqueness of blow-up and blow-down limits provided that the one-homogeneous solution has additional symmetries (integrability through rotations), and establish rigidity at infinity, in the spirit of Simon–Solomon, given suitable conditions on the linearized operator around the one-homogeneous solution. This is joint work with Max Engelstein and Zihui Zhao.