Uniqueness of the Fourier transform on the Euclidean motion group

In this article, we prove that if the Fourier transform of a certain integrable function on the Euclidean motion group is of finite rank, then the function has to vanish identically. Further, we explore a new variance of the uncertainty principle, the Heisenberg uniqueness pairs on the Euclidean motion group as well as on the product group $\mathbb R^n\times K,$ where $K$ is a compact group.