  Scorpius X-1 (Sco X-1) and X-ray transient (XTE) J1751-305 are Low-Mass X-ray
Binaries (LMXBs) that may emit continuous gravitational waves detectable in the
band of ground-based interferometric observatories. Neutron stars in LMXBs
could reach a torque-balance steady-state equilibrium in which angular momentum
addition from infalling matter from the binary companion is balanced by angular
momentum loss, conceivably due to gravitational-wave emission. Torque-balance
predicts a scale for detectable gravitational-wave strain based on observed
X-ray flux. This paper describes a search for Sco X-1 and XTE J1751-305 in LIGO
Science Run 6 data using the TwoSpect algorithm, based on searching for orbital
modulations in the frequency domain. While no detections are claimed, upper
limits on continuous gravitational-wave emission from Sco X-1 are obtained,
spanning gravitational-wave frequencies from 40 to 2040 Hz and projected
semi-major axes from 0.90 to 1.98 light-seconds. These upper limits are
injection validated, equal any previous set in initial LIGO data, and extend
over a broader parameter range. At optimal strain sensitivity, achieved at 165
Hz, the 95% confidence level random-polarization upper limit on dimensionless
strain $h_0$ is approximately $1.8 \times 10^{-24}$. Closest approach to the
torque-balance limit, within a factor of 27, is also at 165 Hz. These are the
first upper limits known to date on $r$-mode emission from this XTE source.
Upper limits are set in particular narrow frequency bands of interest for
J1751-305. The TwoSpect method will be used in upcoming searches of Advanced
LIGO and Virgo data.
