(NLS) equation. Two different peakon equations will be obtained. One is a complex analog of the mCH equation. The other is a peakon analog of the NLS equation itself. Both of these peakon equations are integrable systems, and the main aspects of their integrability will be presented: a Lax pair, a bi-Hamiltonian formulation, a recursion operator, and an infinite hierarchy of symmetries and conservation laws.

These two peakon equations were first derived as 2-component coupled systems in by Lax pair methods, without consideration of the NLS hierarchy. The derivation in the present paper shows how both of these two equations describe negative flows in the NLS hierarchy and provides a common origin for their Lax pairs. In addition, their integrability structure is obtained in a simple, systematic way by combining the tri-Hamiltonian splitting method and the AKNS zero-curvature method [@AblKauNewSeg].

In section \[method\], the derivation of the mCH equation from the mKdV hierarchy will be briefly reviewed, using the tri-Hamiltonian splitting method, which utilizes the bi-Hamiltonian structure of the mKdV hierarchy, and using the recursion operator method, which is closely connected to a zero-curvature matrix formulation of the mKdV hierarchy. Compared to the original presentations of these two methods [@Fuc; 