The flow of granular matter such as sand is often characterized by the motion of
a thin superficial layer near the free surface, while the bulk of the solid remains immobile. A
pair of equations called the BCRE equations (Bouchaud J-P., Cates M. E., Ravi Prakash
J. and Edwards S. F. J. Phys. 4 (1994) 1383) have been proposed to model these flows and
account for the dynamic exchange of mass between moving and stationary grains using the
simplest kinematic considerations. We uncover a new conservation law for the BCRE equations
and its variants that unifies a variety of recent special solutions and show that these equations
support simple waves, and are capable of finite time singularities that correspond to propagating
erosion fronts.