We study the conservative dynamics and stationary configurations of a vortex-antivortex pair in a harmonically trapped two-dimensional Bose-Einstein condensate. We establish the conceptual framework for understanding the stationary states and the topological defect trajectories, through considerations of different mechanisms of vortex motion and the bifurcation of solitonlike stationary solutions. Our insights are based on Lagrangian-based variational calculations, numerical solutions of both the time-dependent and time-independent Gross-Pitaevskii equations, and exact solutions for the noninteracting case.