  A Heisenberg uniqueness pair is a pair $\left(\Gamma, \Lambda\right)$, where
$\Gamma$ is a curve and $\Lambda$ is a set in $\mathbb R^2$ such that whenever
a finite Borel measure $\mu$ having support on $\Gamma$ which is absolutely
continuous with respect to the arc length on $\Gamma$ satisfies
$\hat\mu\vert_\Lambda=0,$ then it is identically $0.$ In this article, we
investigate the Heisenberg uniqueness pairs corresponding to the spiral,
hyperbola, circle and certain exponential curves. Further, we work out a
characterization of the Heisenberg uniqueness pairs corresponding to four
parallel lines. In the latter case, we observe a phenomenon of interlacing of
three trigonometric polynomials.
