  A simple example of an $n$-dimensional admissible complex of planes is given
for the overdetermined $k$-plane transform in $\mathbb{R}^n$. For the
corresponding restricted $k$-plane transform sharp existence conditions are
obtained and explicit inversion formulas are discussed in the general context
of $L^p$ functions. Similar questions are studied for overdetermined Radon type
transforms on the sphere and the hyperbolic space. A theorem describing the
range of the restricted $k$-plane transform on the space of rapidly decreasing
smooth functions is proved.
