  In this paper we consider a four-parameter equation including the
Camassa-Holm and the Dulling-Gottwald-Holm equations, among others. We prove
the existence and uniqueness of solutions to a Cauchy problem involving the
equation using Kato's approach. Conservation laws of the equation up to second
order are also investigated. From these conservation laws we establish some
properties of the solutions of the equation under consideration and we also
find a quadrature for it. The quadrature obtained is of capital importance in a
classification of bounded travelling wave solutions of the equation studied. We
also find some explicit solutions, given in terms of elliptic integrals.
Finally, we classify the members of the equation describing pseudo-spherical
surfaces.
