  We regard the Cauchy problem for a particular Whitham-Boussinesq system
modelling surface waves of an inviscid incompressible fluid layer. The system
can be seen as a weak nonlocal dispersive perturbation of the shallow water
system. The proof of well-posedness relies on energy estimates. However, due to
the symmetry lack of the nonlinear part, in order to close the a priori
estimates one has to modify the traditional energy norm in use. Hamiltonian
conservation provides with global well-posedness at least for small initial
data in the one dimensional settings.
