  Lensed gravitational waves will offer new means to probe the distribution of
matter in the universe, complementary to electromagnetic signals. Lensed
continuous gravitational waves provide new challenges and opportunities beyond
those of transient compact binary coalescence. Here we consider continuous
gravitational waves emitted by isolated neutron stars and lensed by Sgr A$^*$,
the supermassive black hole at the center of our galaxy, a system observable by
the next generation of gravitational wave detectors. We analyze the signatures
of this system in detail, addressing parameter estimation and model selection.
Future detectors can distinguish lensed continuous waves and measure their
parameters with precision $\sim 1 - 10\%$ for sources within $2-4$ Einstein
radii of Sgr A$^*$, depending on the source distance, thanks to the relative
motion of the observer-lens-source system. The chances of observing
strongly-lensed neutron stars increase by one order of magnitude relative to
previous estimates, thanks to the possibility of detecting lensed systems at
several Einstein radii. Multiple images can be resolved with an angular
accuracy $\sim 10$mas, comparable to the best optical telescopes. Image
localization probes deviations from axial symmetry and the existence of
companions to Sgr A$^*$ in regions complementary to stellar orbits and black
hole imaging. Our methods and many of our results extend to other lenses (e.g.
galactic substructure) and sources (e.g. long-lived inspiralling binaries),
rendering lensed continuous gravitational waves into versatile probes of
astrophysics and fundamental physics.
