The effect of highly anisotropic reactivity and interaction between molecules on kinetics of diffusion-influenced liquid-phase reactions is analyzed in detail. The Smoluchowski equation (SE) that describes the space/time evolution of pairs of reacting molecules is solved analytically by the method proposed earlier [A. I. Shushin, Chem. Phys. Lett. 130, 452 (1986); Chem. Phys. 120, 91 (1988)]. The method allows for reducing the original SE in curvilinear translational/rotational coordinates of reacting molecules to the SE in the local Cartesian coordinates in the vicinity of reactive relative orientations. Many formulas for the reaction rate are derived in the realistic models of reactivity using this method. Analysis demonstrates high accuracy of these formulas. The model of reactive patches is shown to give reaction rates that are considerably smaller than those predicted here. Peculiarities of reaction kinetics for molecules with anisotropic reactivity and interaction potential are analyzed in detail. Possible applications of the obtained formulas are briefly discussed.