Situation 1: A test particle of mass m moves around a big mass M in a Keplerian orbit. The orbital period is easily found, given certain initial conditions.
Situation 2: The same system of two bodies M and m, starts interacting under the same initial conditions as in the previous case. However, all of the space now is filled with a uniform distribution of background "dark matter" (density does not depend on position and is not very high, so that recourse to General Relativity is not needed). This background might affect m and M only gravitationally (i.e., there is no dissipation). Will the orbital period of m change?
I encounter a paradox here. Thinking on a primitive level, the mass inside the orbit is increased in the Situation 2 due to the background matter, so the motion must be different now. On the other hand, in the Situation 2 the gravitational potential in each point is changed only by a constant in comparison with Situation 1, so it cannot affect the dynamics, as this constant difference vanishes when taking the gradient to calculate forces.