  We present the analysis of a peculiar W~Virginis (pWVir) type-II Cepheid,
OGLE-LMC-T2CEP-211 ($P_{puls}=9.393\,d$), in a double-lined binary system
($P_{orb}=242\,d$), which shed light on virtually unknown evolutionary status
and structure of pWVir stars. The dynamical mass of the Cepheid (first ever for
a type-II Cepheid) is $0.64\pm{}0.02\,M_\odot$ and the radius
$R=25.1\pm{}0.3\,R_\odot$. The companion is a massive ($5.67\,M_\odot$)
main-sequence star obscured by a disk. Such configuration suggests a mass
transfer in the system history. We found that originally the system
($P_{orb}^{init}=12\,d$) was composed of $3.5$ and $2.8\,M_\odot$ stars, with
the current Cepheid being more massive. The system age is now $\sim{}$200 My,
and the Cepheid is almost completely stripped of hydrogen, with helium mass of
$\sim{}92\%$ of the total mass. It finished transferring the mass 2.5 My ago
and is evolving towards lower temperatures passing through the instability
strip. Comparison with observations indicate a reasonable
$2.7\cdot{}10^{-8}\,M_\odot/y$ mass loss from the Cepheid. The companion is
most probably a Be main-sequence star with $T=22000\,K$ and $R=2.5\,R_\odot$.
Our results yield a good agreement with a pulsation theory model for a
hydrogen-deficient pulsator, confirming the described evolutionary scenario. We
detected a two-ring disk ($R_{disk}\sim\,116\,R_{\odot}$) and a shell
($R_{shell}\sim\,9\,R_{\odot}$) around the companion, that is probably a
combination of the matter from the past mass transfer, the mass being lost by
the Cepheid due to wind and pulsations, and a decretion disk around a rapidly
rotating secondary. Our study together with observational properties of pWVir
stars suggests that their majority are products of a similar binary evolution
interaction.
