Context. The structure of the wind from the cool giants in symbiotic binaries carries important information for understanding the wind mass transfer to their white dwarf companions, its fuelling, and thus the path towards different phases of symbiotic-star evolution. Aims. In this paper, we indicate a non-spherical distribution of the neutral wind zone around the red giant (RG) in the symbiotic binary star, EG And. We concentrate in particular on the wind focusing towards the orbital plane and its asymmetry alongside the orbital motion of the RG. Methods. We achieved this aim by analysing the periodic orbital variations of fluxes and radial velocities of individual components of the Hα and O III λ5007 lines observed on our high-cadence medium (R ∼ 11 000) and high-resolution (R ∼ 38 000) spectra. Results. The asymmetric shaping of the neutral wind zone at the near-orbital-plane region is indicated by: (i) the asymmetric course of the Hα core emission fluxes along the orbit; (ii) the presence of their secondary maximum around the orbital phase φ = 0.1, which is possibly caused by the refraction effect; and (iii) the properties of the Hα broad wing emission originating by Raman scattering on H0 atoms. The wind is substantially compressed from polar directions to the orbital plane as constrained by the location of the O III λ5007 line emission zones in the vicinity of the RG at/around its poles. The corresponding mass-loss rate from the polar regions of ≲10−8 M⊙ yr−1 is a factor of ≳10 lower than the average rate of ≈10−7 M⊙ yr−1 derived from nebular emission of the ionised wind from the RG. Furthermore, it is two orders of magnitude lower than that measured in the near-orbital-plane region from Rayleigh scattering. Conclusions. The startling properties of the nebular O III λ5007 line in EG And provides an independent indication of the wind focusing towards the orbital plane – the key to understanding the efficient wind mass transfer in symbiotic binary stars.