The existence of symmetric wave functions describing Bose-Einstein condensates in a one-dimensional and a fully three-dimensional double-well setup is investigated theoretically. When particles are removed from one well and coherently injected into the other the external potential is symmetric. We solve the underlying Gross-Pitaevskii equation by way of the time-dependent variational principle (TDVP) and show that the symmetry of the external potential is preserved by both the wave functions and the nonlinear Hamiltonian as long as eigenstates with real eigenvalues are obtained. broken solutions of the time-independent Gross-Pitaevskii equation are also found but have no physical relevance. To prove the applicability of the TDVP we compare its results with numerically exact solutions in the one-dimensional case. The linear stability analysis and the temporal evolution of condensate wave functions demonstrate that the symmetric condensates are stable and should be observable in an experiment.