We investigate the existence of solutions to systems of N differential equations representing connections between minima of potentials with several equal depths in ℝn. Using variational techniques and in particular a method introduced by Alikakos and Fusco we first prove such existence for N ≥ 2 and two minima. Dealing next with symmetric potentials corresponding to bulk free energies in crystals, we establish existence for N ≥ 2 in various cases of more than two minima. Finally, we obtain a sufficient condition establishing existence of connections to potentials which are not necessarily symmetric for arbitrary N and three minima.