  Lower order conservation laws and symmetries of a family of hyperbolic
equations having the Camassa-Holm equation as a particular member are obtained.
We show that the equation has two conservation laws with zeroth order
characteristics and that its symmetries are generated by translations in the
independent variables and a certain scaling, as well as some invariant
solutions are studied. Next, we consider persistence and asymptotic properties
for the solutions of the equation considered. In particular, we analyse the
behaviour of the solutions of the equation for large values of the spatial
variable. We show that if the initial data has a certain asymptotic exponential
decaying, then such property persists for any time as long as the solution
exists. Moreover, depending on the behaviour of the initial data for large
values of the spatial variable and if for some further time the solution shares
the same behaviour, then it necessarily vanishes identically. Finally, we prove
unique continuation results for the solutions of the equation.
