  Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy
functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the
half-space $\R^{n} \times [0,+\infty)$ with dilations of graphs of global
minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at
distance 1 from the origin.
