Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

In this article, we prove that (unit sphere, non-harmonic cone) is a Heisenberg uniqueness pair for the symplectic Fourier transform on C n . We derive spheres as well cones are determining sets spectral projections of finite measure supported unit sphere. Further, if finitely function step two nilpotent Lie group arbitrary rank, then must be zero. The latter result correlates to annihilating Weyl transform.