  Unique continuation properties for a class of evolution equations defined on
Banach spaces are considered from two different point of views: the first one
is based on the existence of conserved quantities, which very often translates
into the conservation of some norm of the solutions of the system in a suitable
Banach space. The second one is regarded to well-posed problems. Our results
are then applied to some equations, most of them describing physical processes
like wave propagation, hydrodynamics, and integrable systems, such as the $b-$;
Fornberg-Whitham; potential and $\pi-$Camassa-Holm; generalised Boussinesq
equations; and the modified Euler-Poisson system.
