  Two exact evaluation formulae for multiple rarefied elliptic beta integrals
related to the simplest lens space are proved. They generalize evaluations of
the type I and II elliptic beta integrals attached to the root system $C_n$. In
a special $n=1$ case, the simplest $p\to 0$ limit is shown to lead to a new
class of $q$-hypergeometric identities. Symmetries of a rarefied elliptic
analogue of the Euler-Gauss hypergeometric function are described and the
respective generalization of the hypergeometric equation is constructed. Some
extensions of the latter function to $C_n$ and $A_n$ root systems and
corresponding symmetry transformations are considered. An application of the
rarefied type II $C_n$ elliptic hypergeometric function to some eigenvalue
problems is briefly discussed.
