  A unique signature of the modified Newtonian dynamics (MOND) paradigm is its
peculiar behavior in the vicinity of the points where the total Newtonian
acceleration exactly cancels. In the Solar System, these are the saddle points
of the gravitational potential near the planets. Typically, such points are
embedded into low-acceleration bubbles where modified gravity theories a` la
MOND predict significant deviations from Newton's laws. As has been pointed out
recently, the Earth-Sun bubble may be visited by the LISA Pathfinder spacecraft
in the near future, providing a unique occasion to put these theories to a
direct test. In this work, we present a high-precision model of the Solar
System's gravitational potential to determine accurate positions and motions of
these saddle points and study the predicted dynamical anomalies within the
framework of quasi-linear MOND. Considering the expected sensitivity of the
LISA Pathfinder probe, we argue that interpolation functions which exhibit a
"faster" transition between the two dynamical regimes have a good chance of
surviving a null result. An example of such a function is the QMOND analog of
the so-called simple interpolating function which agrees well with much of the
extragalactic phenomenology. We have also discovered that several of Saturn's
outermost satellites periodically intersect the Saturn-Sun bubble, providing
the first example of Solar System objects that regularly undergo the MOND
regime.
