Cylindrical symmetry for solutions to equations on the Heisenberg group

This paper investigates the cylindrical symmetry of solutions to fractional equations on the Heisenberg group, extending classical symmetry results in Euclidean spaces to the sub-Riemannian framework. Key challenges arising from non-commutative geometry, such as anisotropic scaling and horizontal gradients, are addressed. Using the method of moving planes adapted to the Heisenberg group structure, we establish symmetry and monotonicity properties for positive solutions under specific assumptions on nonlinear terms.