  We define a parametric Radon transform $R$ that assigns to a Sobolev function
on the cylinder $\mathbb{S}\times \mathbb{R}$ in $\mathbb{R}^3$ its mean values
along sets $E_\zeta$ formed by the intersections of planes through the origin
and the cylinder. We show that $R$ is a continuous operator, prove an inversion
formula, provide a support theorem, as well as a characterization of its null
space. We conclude by presenting a formula for the dual transform $R^*$. We
show that $R$ and its dual $R^*$ are related to the right-sided and left-sided
Chebyshev fractional integrals. Using this relationship, we characterize the
null space of $R$ and $R^*$ and provide an inversion formula for $R^*$.
