  In the present study we consider three two-component (integrable and
non-integrable) systems which describe the propagation of shallow water waves
on a constant shear current. Namely, we consider the two-component Camassa-Holm
equations, the Zakharov-Ito system and the Kaup--Boussinesq equations all
including constant vorticity effects. We analyze both solitary and
periodic-type travelling waves using the simple and geometrically intuitive
phase space analysis. We get the pulse-type solitary wave solutions and the
front solitary wave solutions. For the Zakharov-Ito system we underline the
occurrence of the pulse and anti-pulse solutions. The front wave solutions
decay algebraically in the far field. For the Kaup-Boussinesq system,
interesting analytical multi-pulsed travelling wave solutions are found.
