  We outline a new method suggested by Conway (2016) for solving the two-body
problem for solid bodies of spheroidal or ellipsoidal shape. The method is
based on integrating the gravitational potential of one body over the surface
of the other body. When the gravitational potential can be analytically
expressed (as for spheroids or ellipsoids), the gravitational force and mutual
gravitational potential can be formulated as a surface integral instead of a
volume integral, and solved numerically. If the two bodies are infinitely thin
disks, the surface integral has an analytical solution. The method is exact as
the force and mutual potential appear in closed-form expressions, and does not
involve series expansions with subsequent truncation errors. In order to test
the method, we solve the equations of motion in an inertial frame, and run
simulations with two spheroids and two infinitely thin disks, restricted to
torque-free planar motion. The resulting trajectories display precession
patterns typical for non-Keplerian potentials. We follow the conservation of
energy and orbital angular momentum, and also investigate how the spheroid
model approaches the two cases where the surface integral can be solved
analytically, i.e. for point masses and infinitely thin disks.
