Two-body problem, but with one of the bodies fixed

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The two-body problem discusses the orbit of two celestial objects (planets) around each other, often with one of them having heavier mass than the other, which has smaller mass.

If the heavier mass is instead treated as a fixed center of force, and the motion of the lighter mass may be solved for directly by one-body methods, how do the maths of the two-body problem change to accommodate the new assumption of one of the masses being stationary/immobile? or how can that be accounted for in the formula?

Can't understand the following tutorial's language, and I think he's overdoing it

Reduction of two body problem into one body problem