  New light scalar degrees of freedom may alleviate the dark matter and dark
energy problems, but if coupled to matter, they generally mediate a fifth
force. In order for this fifth force to be consistent with existing
constraints, it must be suppressed close to matter sources, e.g. through a
non-linear screening mechanism. In this work, we investigate the
non-relativistic two-body problem in shift-symmetric scalar-tensor theories
that exhibit kinetic screening ($k$-mouflage), both numerically and
analytically. We develop an approximate scheme, based on a Hodge-Helmholtz
decomposition of the Noether current associated to the shift symmetry, allowing
for a qualitative insight into the dynamics and yielding results in good
agreement with the numerical ones in most of the parameter space. We apply the
formalism to polynomial $k$-essence and to Dirac-Born-Infeld (DBI) type
theories, as well as to theories that develop ``anti-screening''. In the deep
nonlinear regime, we find that the fifth force is screened slightly more
efficiently in equal-mass systems than in extreme mass-ratio ones. However, we
find that systems with comparable masses also exhibit regions where the
screening is ineffective. These descreened spheroidal regions (bubbles) could
in principle be probed in the solar system with sufficiently precise space
accelerometers.
