  Two integrable $U(1)$-invariant peakon equations are derived from the NLS
hierarchy through the tri-Hamiltonian splitting method. A Lax pair, a recursion
operator, a bi-Hamiltonian formulation, and a hierarchy of symmetries and
conservation laws are obtained for both peakon equations. These equations are
also shown to arise as potential flows in the NLS hierarchy by applying the NLS
recursion operator to flows generated by space translations and $U(1)$-phase
rotations on a potential variable. Solutions for both equations are derived
using a peakon ansatz combined with an oscillatory temporal phase. This yields
the first known example of a peakon breather. Spatially periodic counterparts
of these solutions are also obtained.
