The gravitational potential $\Phi$ of an infinite rod in newtonian gravity is $\Phi \sim \ln(r)$. This is the same as the gravitational potential of a point charge in two-dimensional Newtonian gravity (see https://en.wikipedia.org/wiki/Newtonian_potential). They are the same, because both systems exhibit cylindrical symmetry and Gauss Law yields a logarithmic potential in this case.
In general relativity the solution for a static cylindrical spacetime is the Levi-Civita spacetime, which in the Newtonian limit will also give a potential $\Phi \sim \ln(r)$ (see e.g. https://arxiv.org/abs/1901.06561).
But what I can't understand is that in (2+1) dimensional general relativity it is said the spacetime is flat outside of a mass point, so in the Newtonian limit $\Phi \sim 0$. This is claimed despite authors stating that a point particle in general relativity in (2+1) dimensions is equivalent to a (infinite) string in (3+1) dimensions.
"We discussed the global properties of the (locally flat) geometries generated by moving point particles in 2+1 dimensions, or equivalently by parallel moving cosmic strings in 3+1 dimensions." (Deser, Jackiw, t'Hooft (1992))
"There is also a close relation to cosmic strings in four dimensions since the space-time of an infinite straight string is effectively three-dimensional." (Deser, Jackiw (1988))
So why is there a difference between Newtonian Potential derived from Levi-Civita spacetime and the Newtonian potential derived from a (2+1) dimensional general relativity? Is general relativity in (2+1) dimensions not simply a cross section through a cylindrical symmetric spacetime in (3+1) dimensions? What is it then?