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Since $(\mathbb{H}^n\rtimes U(n),U(n))$ is a Gelfand pair, an exact analogueof the Heisenberg group result due to Narayanan and Ratnakumar is not possiblefor the Heisenberg motion group. In this article, we prove that if the Weyltransform of a finitely supported integrable function on the Heisenberg motiongroup is non-zero only for finitely many Fourier-Wigner pieces and have finiterank, then the function must be zero. We also prove an analogue of theHeisenberg group result on the quaternion Heisenberg group. In the end, aquantitative interpretation of these results is described through strongannihilating pair for the Weyl transform.

Benedicks-Amrein-Berthier theorem for the Heisenberg motion group and quaternion Heisenberg group