  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+\Delta T $ 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 $ \nu_{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 $
\nu=\nu_{ij}/(1+z) $. For different $z$ (and, consequently, for different $ \nu
$), 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 $ \Delta T/T $, the luminescent lines can be very
noticeable in the spectrum of blackbody background radiation.
