Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom

We prove in this note the local (in time) well-posedness of a broad class of $2\times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond, Dutykh and Mitsotakis. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.