We consider partial symmetry of the least energy solutions of some nonlinear Schrödinger systems. We show that in the attractive case, all components of the least energy solutions possess foliated Schwarz symmetry with respect to the same point.
Wang, Z.-Q., & Willem, M. (2014). Partial symmetry of vector solutions for elliptic systems. Journal d’Analyse Mathematique, 122(1), 69-85. https://doi.org/10.1007/s11854-014-0003-z (Original work published 2014)