  Based on Newtonian dynamics, observations show that the luminous masses of
astrophysical objects that are the size of a galaxy or larger are not enough to
generate the measured motions which they supposedly determine. This is
typically attributed to the existence of dark matter, which possesses mass but
does not radiate (or absorb radiation). Alternatively, the mismatch can be
explained if the underlying dynamics is not Newtonian. Within this conceptual
scheme, Modified Newtonian Dynamics (MOND) is a successful theoretical
paradigm. MOND is usually expressed in terms of a nonlinear Poisson equation,
which is difficult to analyse for arbitrary matter distributions. We study the
MONDian gravitational field generated by slightly non-spherically symmetric
mass distributions based on the fact that both Newtonian and MONDian fields are
conservative (which we refer to as the compatibility condition). As the
non-relativistic version of MOND has two different formulations (AQUAL and
QuMOND) and the compatibility condition can be expressed in two ways, there are
four approaches to the problem in total. The method involves solving a suitably
defined linear deformation potential, which generally depends on the choice of
MOND interpolation function. However, for some specific form of the deformation
potential, the solution is independent of the interpolation function.
