  Lebesgue space estimates are obtained for the circular maximal function on
the Heisenberg group $\mathbb{H}^1$ restricted to a class of Heisenberg radial
functions. Under this assumption, the problem reduces to studying a maximal
operator on the Euclidean plane. This operator has a number of interesting
features: it is associated to a non-smooth curve distribution and, furthermore,
fails both the usual rotational curvature and cinematic curvature conditions.
