Finiteness of kite relative equilibria in the five-vortex and five-body problems

We study the finiteness of planar relative equilibria of the Newtonian fivebody problem and in the five-vortex problem in the case that configurations form a symmetric kite (three points on a line and two additional points placed symmetrically with respect to that line). We can prove that the equivalence classes of such relative equilibria are finite with some possible exceptional cases. These exceptional cases are given explicitly as polynomials in the masses (or vorticities in the vortex problem). These results depend on computations performed with the software Sage, Singular, Magma, and Gfan.