  We study a regularization of the classical Saint-Venant (shallow-water)
equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci.
Numer. Simulat. 55 (2018) 237-247). This regularization is non-dispersive and
formally conserves mass, momentum and energy. We show that for every classical
shock wave, the system admits a corresponding non-oscillatory traveling wave
solution which is continuous and piecewise smooth, having a weak singularity at
a single point where energy is dissipated as it is for the classical shock. The
system also admits cusped solitary waves of both elevation and depression.
