In this work,we revise the classical Maxwells constitutive relation and formulate a system of first-order partial differential equations with two parameters for compressible viscoelastic fluid flows.The system is shown to possess a nice conservation-dissipation(relaxation)structure and therefore is symmetrizable hyperbolic.Moreover,for smooth flows we rigorously verify that the revised Maxwells constitutive relations are compatible with Newtons law of viscosity.