Abstract: This work continues a series of studies by the authors, where a comparative analysis of the influence of topography, β-effect, and gradient variability of a background flow on the propagation of barotropic topographic Rossby waves is carried out. The novelty of this study lies in the fact that here we consider a non-zonal shear flow and also non-zonal topographic changes. We establish that the non-zonal shear flow enhances the β-effect from the north side of the main flow and reduces from the south side. Contrary, the influence of the joint effect of the Earth rotation and bottom topography enhances the β-effect in the study area and the joint effect of the shear flow and topography reduces it slightly. We exclude the effect of stratification based on the results already obtained that short waves are not observed practically, and also the effect of stratification for Rossby long waves is insignificant. The transverse variability of the shear plane-parallel flow in the WKB approximation is studied. This allows us to obtain a dispersion relation for the flat barotropic topographic Rossby waves with taking into account the above conditions. The estimates of the dispersion equation terms are obtained for the Kuroshio Extension area, where a flow branch turns to the northeast and makes an angle of 55° with latitude. Applied the exponential approximation for bottom topography we obtain that the First approximation by Rhines for the Rossby waves is true for the study area because the parameter of the relief transverse change l₀ equals 780 km (exceeds typical Rossby wavelengths). It allows simplifying the corresponding dispersion relation.