admit a real reduction, while the other peakon equation has a complex mCH-type form whose real reduction is given by the mCH equation.

In section \[laxpair\], for each of the two $U(1)$-invariant peakon equations, their bi-Hamiltonian structure will be used to derive a hierarchy of symmetries and conservation laws, and a Lax pair will be obtained by modifying the zero-curvature equation of the mCH equation.

In section \[peakons\], solutions for the two $U(1)$-invariant peakon equations will be derived by using a peaked travelling wave expression modified by a temporal phase oscillation, $u=a \exp(\inum(\phi+\omega t)-|x-ct|)$. Specifically, the complex mCH-type equation will be shown to possess only a standard peakon with a constant phase, while in contrast the NLS-type peakon equation will be shown to possess a stationary peakon with a temporal phase oscillation given by $w=\tfrac{1}{3}a^3$. This provides the first ever example of a peakon breather. Moreover, spatially periodic counterparts of these peakons will be derived.

Finally, some concluding remarks will be made in section \[remarks\].

Derivation of the mCH peakon equation {#method}
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We start by considering the mKdV equation $$\label{mkdv-eqn}
v_t + \tfrac{3}{2}v^2 v_x +v_{xxx} =0$$ where we have chosen the scaling factor in the nonlinear term to simplify 