  We report the discovery of Balmer broad absorption lines (BALs) in the quasar
LBQS 1206+1052 and present a detailed analysis of the peculiar absorption line
spectrum. Besides Mg II $\lambda \lambda 2796, 2803$ doublet, BALs are also
detected in He I* multiplet at $\lambda \lambda 2946, 3189, 3889$ \AA arising
from metastable helium $2^3S$ level, and in H$\alpha$ and H$\beta$ from excited
hydrogen H I* $n=2$ level, which are rarely seen in quasar spectra. We identify
two components in the BAL troughs of $\Delta v\sim$2000 km s$^{-1}$ width: One
component shows an identical profile in H I*, \hei* and \mgii with its centroid
blueshifted by $-v_{\rm c}\approx 726$ km\ s$^{-1}$. The other component is
detected in \hei* and \mgii with $-v_{\rm c}\approx 1412$ km s$^{-1}$. We
estimate the column densities of H I*, He I*, and Mg II, and compare them with
possible level population mechanisms. Our results favor the scenario that the
Balmer BALs originate in a partially ionized region with a column density of
$N_{\rm H}\sim 10^{21-22}$ cm$^{-2}$ for an electron density of $n_e\sim
10^{6-8} $cm$^{-3}$ via Ly$\alpha$ resonant scattering pumping. The harsh
conditions needed may help to explain the rarity of Balmer absorption line
systems in quasar spectra. With an $i$-band PSF magnitude of 16.50, LBQS
1206+1052 is the brightest Balmer-BAL quasar ever reported. Its high brightness
and unique spectral properties make LBQS 1206+1052 a promising candidate for
follow-up high-resolution spectroscopy, multi-band observations, and long-term
monitoring.
