  It is well known that a line in $\mathbb R^2$ is not a set of injectivity for
the spherical means for odd functions about that line. We prove that any line
passing through the origin is a set of injectivity for the twisted spherical
means (TSM) for functions $f\in L^2(\mathbb C),$ whose each of spectral
projection $ e^{\frac{1}{4}|z|^2}f\times\varphi_k$ is a polynomial. Then, we
prove that any Coxeter system of even number of lines is a set of injectivity
for the TSM for $L^q(\mathbb C),~1\leq q\leq2.$
