  Ultra-compact X-ray binaries (UCXBs) are low-mass X-ray binaries with
ultra-short orbital periods (usually less than 1 hour) and hydrogen-poor donor
stars, which are proposed to be the potential LISA sources. In this work, we
firstly employ the MESA code to examine the parameter space of the progenitors
of UCXBs that LISA will detect. Our simulations indicate that the initial
binaries with a neutron star and a $0.4-3.3~M_{\odot}$ companion star in an
orbit of initial orbital period smaller than the bifurcation period could
evolve into UCXBs, some of which will emit gravitational wave signals that can
be detectable by the LISA. However, the initial orbital periods of the binaries
that will evolve into UCXB-LISA sources in a distance of 10 kpc are located in
a very narrow range, i.e. the formation of these LISA source requires an
extremely fine-tuning of initial parameter. According to the characteristic
stains and the derived maximum detectable distances, four sources among eight
UCXBs with the observed distances are expected to be detected by the LISA.
Based on the parameter space given by the detailed binary evolution models and
the rapid binary star evolution code, the birthrate of UCXBs appearing as LISA
sources in the Galaxy is estimated to be $(2-2.6)\times10^{\rm -6}~\rm
yr^{-1}$. Considering the contribution of UCXBs in the globular clusters, the
number of UCXB-LISA sources can reach $240 - 320$. Although the formation
condition is severe, the detectability of UCXBs by the LISA is still optimistic
and significant because they provide an opportunity to pursue full
multi-messenger investigations.
