We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831] for Maxwell fluids to a class of systems that define unequivocally multi-dimensional visco-elastic flows. Precisely, within a general setting for continuum mechanics, we specify constitutive assumptions i) that ensure the unequivocal definition of motions satisfying widely-admitted physical principles, and ii) that contain [ESAIM:M2AN 55 (2021) 807-831] as one particular realization of those assumptions. The new class can capture the mechanics of various materials, from solids to viscous fluids, possibly with temperature dependence and heat conduction.