How would gravity work in this alternate universe?

In my universe, gravity varies with the inverse of the distance (instead of the inverse square of the distance). I already know that this means that all circular orbits around the same object will have the same speed, and no true escape trajectories can exist, but there are some facts that I can't figure out about how the universe will work.

What shape will non-circular orbits take (or do they not exist)? Also, is there anything else weird about gravity in this universe?

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How would gravity work in this alternate universe?

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