  Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a
finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$. An
important branching problem is to determine the finite-dimensional
highest-weight $\mathfrak{g}$-modules which admit a $\mathfrak{k}$-coinvariant,
and thus appear as functions in a corresponding supersymmetric space
$\mathcal{G}/\mathcal{K}$. This is the super-analogue of the Cartan-Helgason
theorem. We solve this problem via a rank one reduction and an understanding of
reflections in singular roots, which generalize odd reflections in the theory
of Kac-Moody superalgebras. An explicit presentation of spherical weights is
provided for every pair when $\mathfrak{g}$ is indecomposable.
