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The study is devoted to investigation of the liquid rheology effect on low-frequency wave propagation in a gap between two elastic cylindrical shells, filled with viscoelastic polymeric solution. The presence of free gas traces in liquid, typical for polymeric solutions, is accounted for. It is supposed that volume concentration of free gas is small, the microbubbles are size distributed, and the losses in the wave stem from both liquid-shell and liquid-bubble dynamic interaction. Rheology of polymeric liquid is described by generalized Maxwell model with Newtonian viscous term, responsible for low-molecular solvent contribution in the stress tensor. Dynamic equations for thin elastic shells are formulated within Kirchhoff-Love approximation; hydrodynamics of liquid flow in the gap in the wave is described using quasi-one-dimensional approach. The dispersion equation, accounting for structure coupling in the wave (both liquid-solid and liquid-gas) is obtained and studied numerically for different parameter values. The analysis has revealed strong dependence of sound dispersion in the system from liquid properties, gas concentration, shell elasticity and the gap width and has showed important effect of structure coupling on the wave propagation. Results of the study may be useful for modeling of dynamic behavior of structure elements at polymer extrusion and in free gas monitoring at polymer processing.