  In this article, we prove that any pair of perpendicular planes is a set of
injectivity for the twisted spherical means (TSM) for $L^p(\mathbb C^n)
(n\geq2)$ with $1\leq p\leq2.$ Then, we imitate that any Coxeter system of even
number of planes is a set of injectivity for the TSM for $L^p(\mathbb C^n).$ We
further observe that a set $S_R^{2n-1}\times\mathbb C$ is a set of injectivity
for the TSM for a ceratin class of functions on $\mathbb C^{n+1}.$
