In this paper we study positive solutions to the CR Yamabe equation on groups of Heisenberg type. For the subclass of groups of Iwasawa type, we characterize those solutions that have partial symmetry, that is, those that are invariant with respect to the action of the orthogonal group in the first layer of the Lie algebra.
Nicola Garofalo. Dimiter Vassilev. "Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type." Duke Math. J. 106 (3) 411 - 448, 15 February 2001. https://doi.org/10.1215/S0012-7094-01-10631-5