  Our aim in the present work is to identify all the possible standing wave
configurations involving few vortices of different charges in an atomic
Bose-Einstein condensate (BEC). In this effort, we deploy the use of a
computational algebra approach in order to identify stationary multi-vortex
states with up to 6 vortices. The use of invariants and symmetries enables
deducing a set of equations in elementary symmetric polynomials, which can then
be fully solved via computational algebra packages within Maple. We retrieve a
number of previously identified configurations, including collinear ones and
polygonal (e.g. quadrupolar and hexagonal) ones. However, importantly, we also
retrieve a configuration with 4 positive charges and 2 negative ones which is
unprecedented, to the best of our knowledge, in BEC studies. We corroborate
these predictions via numerical computations in the fully two-dimensional PDE
system of the Gross-Pitaevskii type which characterizes the BEC at the
mean-field level.
