  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.
