Aims. A model of heliosheath density and energy spectra of alpha-particles and He+ ions carried by the solar wind is developed. Neutralization of heliosheath He+ ions, mainly by charge exchange (CX) with neutral interstellar H and He atoms, gives rise to ~0.2 - ~100 keV fluxes of energetic neutral He atoms (He ENA). Such fluxes, if observed, would give information about plasmas in the heliosheath and heliospheric tail. Methods. Helium ions crossing the termination shock (TS) constitute suprathermal (test) particles convected by (locally also diffusing through) hydrodynamically calculated background plasma flows (three versions of flows are employed). The He ions proceed from the TS towards heliopause (HP) and finally to the heliospheric tail (HT). Calculations of the evolution of alpha- and He+ particle densities and energy spectra include binary interactions with background plasma and interstellar atoms, adiabatic heating (cooling) resulting from flow compression (rarefaction), and Coulomb scattering on background plasma. Results. Neutralization of suprathermal He ions leads to the emergence of He ENA fluxes with energy spectra modified by the Compton-Getting effect at emission and ENA loss during flight to the Sun. Energy-integrated He ENA intensities are in the range ~0.05

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