A constrained theological model for polymer solutions and melts is developed using the GENERIC formulation. The constraint, det c = constant, where c is a conformation tensor, is introduced in the general formulation using a modified mobility tenser, Lambda(1), and a Helmholtz free-energy function expressed in terms of the scaled invariants of the conformation tenser. The predictions of this family of＇ "volume-preserving" models are illustrated for a modified Hookean energy function and several expressions of the modified mobility tenser. The predictions are compared to experimental data taken from the literature for polymer melts and polymer solutions. The sensitivity of the predictions to model parameters (a maximum of four fdr this particular case) shows that in steady and transient shear flows, it is possible to cover a wide range of theological behaviors generally observed for polymer melts and solutions.;Comparison with experimental results for polymer melts and solutions shows that the model predicts very well the steady shear viscosity and first normal stress coefficient behavior for an extended range of shear rates. Model predictions for material functions in shear start-up experiments compare fairly well with the experimental data. The modified Hookean energy function proposed to illustrate the approach is found to give larger overshoots than those observed experimentally in sheer start-up experiments. Predictions for shear stress relaxation failed to compare with the set of experimental data for which this property is available. It is believed that these drawbacks with this particular choice of the Helmholtz free-energy function can be overcome using multiple conformation variables. This can be easily done in this formulation.