A result of Korányi that describes the structure of the space of polynomial solutions to the Heisenberg Laplacian operator is generalized to the canonical central systems on the block Heisenberg groups. These systems of differential operators generalize the Heisenberg Laplacian and, like it, admit large algebras of conformal symmetries. The main result implies that in most cases all polynomial solutions can be obtained from a single one by the repeated application of conformal symmetry operators.
Anthony C. Kable. "The structure of the space of polynomial solutions to the canonical central systems of differential equations on the block Heisenberg groups: A generalization of a theorem of Korányi." Tohoku Math. J. (2) 70 (4) 523 - 545, 2018. https://doi.org/10.2748/tmj/1546570824