Non-harmonic cones are Heisenberg uniqueness pairs for the Fourier transform on ${\mathbb{R}}^n$

In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on ${\mathbb{R}}^n\mathrm{.}$ We derive that $(S^2,\text{ paraboloid})$ and $\left(S^2, \text{ geodesic of } S_r(o)\right)$ are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in ${\mathbb{R}}^3\mathrm{.}$ Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.