We propose in this work the first symmetric hyperbolic system of conservation laws to describe viscoelastic flows of Maxwell fluids, i.e. fluids with memory that are characterized by one relaxation-time parameter. Precisely, the system of quasilinear PDEs is detailed for the shallow- water regime, i.e. for hydrostatic incompressible 2D flows with free surface under gravity. It generalizes Saint-Venant system to viscoelastic flows of Maxwell fluids, and encompasses previous works with F. Bouchut. It also generalizes the (thin-layer) elastodynamics of hyperelastic materials to viscous fluids, and to various rheologies between solid and liquid states that can be formulated using our new variable as material parameter. The new viscoelastic flow model has many potential applications, additionally to falling into the theoretical framework of (symmetric hyper- bolic) systems of conservation laws. In computational rheology, it offers a new approach to the High-Weissenberg Number Problem (HWNP). For transient geophysical flows, it offers perspectives of thermodynamically-compatible numerical simulations, with a Finite-Volume (FV) discretization say. Besides, one FV discretization of the new continuum model is proposed herein to precise our ideas incl. the physical meaning of the solutions. Perspectives are finally listed after some numerical simulations.