We have considered the formation of the luminescent subordinate HeI lines by the absorption of continuum radiation from a source in the lines of the main HeI series in the expanding Universe. It is suggested that at some moment of time, corresponding to the redshift z ₀, a burst of superequilibrium blackbody radiation with a temperature T? +? ΔT occurs. This radiation is partially absorbed at different z? < ? z ₀ in the lines of the main HeI series and then converted into the radiation of subordinate lines. If ν _ij is the laboratory frequency of the transition of some subordinate line originating at some z, then in the present time its frequency will be ν? =? ν _ij/(1? +? z). For different z (and, consequently, for different ν), the quantum yield for the subordinate lines of para- and orthohelium - the number of photons emitted in the subordinate line, per one initial excited atom and line profiles are calculated. Different pumping channels were considered. Spatial and angular distributions of radiation intensity of luminescent lines for the spherically symmetric radiation sources are presented. It is shown that for sufficiently large ΔT/T, the luminescent lines can be very noticeable in the spectrum of blackbody background radiation.