  We consider a Lax pair found by Xia, Qiao and Zhou for a family of
two-component analogues of the Camassa-Holm equation, including an arbitrary
function $H$, and show that this apparent freedom can be removed via a
combination of a reciprocal transformation and a gauge transformation, which
reduces the system to triangular form. The resulting triangular system may or
may not be integrable, depending on the choice of $H$. In addition, we apply
the formal series approach of Dubrovin and Zhang to show that scalar equations
of Camassa-Holm type with homogeneous nonlinear terms of degree greater than
three are not integrable.
