Annular solutions to the partitioning problem in a ball

For any $n\in \mathbb{N}$, $n\ge 2$, we construct a real analytic, one-parameter family of compact embedded CMC annuli with free boundary in the unit ball ${\mathbb{B}}^3$ of ${\mathbb{R}}^3$ with a prismatic symmetry group of order $4n$. These examples give a negative answer to the uniqueness problem by Nitsche and Wente of whether any annular solution to the partitioning problem in the ball should be rotational.