  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 $\mathbb R^n.$ We
derive that $\left(S^2, \text{ paraboloid}\right)$ 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 $\mathbb R^3.$ Further, we correlate
the problem of Heisenberg uniqueness pairs to the sets of injectivity for the
spherical mean operator.
