We consider singular solutions of a system of two cross-coupled Camassa-Holm
(CCCH) equations. This CCCH system admits peakon solutions, but it is not in
the two-component CH integrable hierarchy. The system is a pair of coupled
Hamiltonian partial differential equations for two types of solutions on the
real line, each of which separately possesses exp(-|x|) peakon solutions with a
discontinuity in the first derivative at the peak. However, there are no
self-interactions, so each of the two types of peakon solutions moves only
under the induced velocity of the other type. We analyse the `waltzing'
solution behaviour of the cases with a single bound peakon pair (a peakon
couple), as well as the over-taking collisions of peakon couples and the
antisymmetric case of the head-on collision of a peakon couple and a peakon
anti-couple. We then present numerical solutions of these collisions, which are
inelastic because the waltzing peakon couples each possess an internal degree
of freedom corresponding to their `tempo' -- that is, the period at which the
two peakons of opposite type in the couple cycle around each other in phase
space. Finally, we discuss compacton couple solutions of the cross-coupled
Euler-Poincar\'e (CCEP) equations and illustrate the same types of collisions
as for peakon couples, with triangular and parabolic compacton couples. We
finish with a number of outstanding questions and challenges remaining for
understanding couple dynamics of the CCCH and CCEP equations
