  A new class of neutron stars (NSs) -- hot rapidly rotating non-accreting NSs,
which we propose to call HOFNARs (HOt and Fast Non-Accreting Rotators) or "hot
widows" (in analogy with "black widow" pulsars) -- is suggested. We argue that
such stars should originate from the low-mass X-ray binaries (LMXBs) provided
that they were unstable with respect to excitation of $r$-modes at the end of
accretion epoch (when their low-mass companions ceased to fill the Roche lobe).
High temperature of "hot widows"/HOFNARs is maintained by $r$-mode dissipation
rather than by accretion. We analyse observational properties of "hot
widows"/HOFNARs and demonstrate that these objects form a specific separate
class of neutron stars. In particular, some of the most stable X-ray sources
among the candidates to quiescent LMXB systems (qLMXBs), can, in fact, belong
to that new class. We formulate observational criteria which allow to
distinguish "hot widows"/HOFNARs from qLMXB systems, and argue that available
observations of X-ray sources 47 Tuc X5 and X7 satisfy (or, at least, do not
contradict) these criteria. In addition, we discuss indirect evidences in favor
of "hot widows"/HOFNARs existence, following from the analysis of observations
and predictions of population synthesis theories. If that new class of NSs does
exist, it would prove the possibility to emit gravitational waves by
mass-current multipole. Various applications of our results, such as prospects
for constraining superdense matter properties with hot widows"/HOFNARs, are
analyzed.
