In 1984, Benjamin proposed a Hamiltonian structure for the Boussinesq model in the two dimensional context. The model describes the motion of an incompressible inviscid fluid with non uniform density, and is used to study internal wave phenomena. The Hamiltonian structure he derived has a peculiarity: in the presence of rigid wall delimiting the fluid domain, it breaks down if the density is not constant along them. This singular behaviour has been investigated later on by Camassa et al. (2014). In that study it is pointed out that the topology of initial conditions can affect the set of conserved quantities of the system.