I learnt that the curvature tensor in 2+1D spacetime is zero in vacuum. How is it possible to come from there to the Newton's theory in 2D + time, where I guess, the gravitational force law is still possible in a for instance solar system made of disks.
If there is no curvature in empty space, the particles (planets) would move on straight lines (at least in an appropriate chosen coordinate system), how would that be compatible with Newton's force law?
The 2+1 dimensional versions of Newtonian gravity and of General Relativity do not match. While you can still interpret Newton's force law as a regular force in two spatial dimensions, 2+1 dimensional GR predicts gravity does not act at a distance.
Einstein gravity in 2+1D is a topological field theory. The linearized EFE is without propagating physical degrees of freedom.
There are typically gravistatic effects from conical singularities caused by point masses. In particular there is no meaningful notion of a Newtonian asymptotic region due to the presence of a deficit angle. Perhaps surprisingly, the Newtonian $ln$ potential has no analog in GR!
See also e.g. this related Phys.SE post.
Try to write the metric: $d\tau^2 = A(r)dt^2 - B(r)dr^2 - r^2d\theta^2$, where it is supposed a circular symmetry around the origin, and a metric with no time dependence. After calculating the connections and the components of the Ricci tensor as a function of A, B and r, suppose a vacuum solution ($R_{\mu \nu} = 0$). The solution of the system of differential equations shows that A and B are constants.
The geodesic equations for this metric show: $\frac{dt}{d\tau} =$ const. $\frac{d^2r}{d\tau^2}$ and $\frac{d^2\theta}{d\tau^2}$ are the usual for plane polar coordinates. So instead of a Schwartschild-like space-time as could be expected (it results from the same procedure for 3+1), the result is a kind of Minkowski space-time, showing no sign of gravity.
