  We present a geometrical demonstration for persistence properties for a
bi-Hamiltonian system modelling waves in a shallow water regime. Both periodic
and non-periodic cases are considered and a key ingredient in our approach is
one of the Hamiltonians of the system. As a consequence of our developments we
improve recent works dealing with unique continuation properties for shallow
water equations, as well as we provide a novel way to prove that the unique
compactly supported solution of the system is necessarily the zero function.
