# The pair potential different that $r^{-1}$

As I know, the potential between two particles of the form ~ $$r^{-1}$$ ($$r$$ is distance between particles) is special, because it solves the Poisson's equation in 3D.
My question is: If I consider for example pair potentials like $$r^{-2}$$, or $$\ln(r)$$, can I construct linear equations for this fields, equivalent to Poisson equation for gravitational potential?

1. If you take E&M you would knew that the solution came with basis in $$r^a$$ where $$a$$ to be integers.
2. For solutions of specific question(such as comet or charges in free space, if I remembered correctly), sometimes $$r^{-1}$$ marked a "tipping-point", of which for $$a<-1$$ they($$r^a$$s) usually satisfy some nice properties. However, if the question get changed, the specialty of $$r^{-1}$$ may no longer holds.
3. Usually, one don't need to consider $$ln(r)$$ when using basis such as $$r^{a}$$s.