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