  We propose a new bilinear Hirota equation for $\tau$-functions associated
with the $E_8$ root lattice, that provides a "lens" generalisation of the
$\tau$-functions for the elliptic discrete Painlev\'e equation. Our equations
are characterized by a positive integer $r$ in addition to the usual elliptic
parameters, and involve a mixture of continuous variables with additional
discrete variables, the latter taking values on the $E_8$ root lattice. We
construct explicit $W(E_7)$-invariant hypergeometric solutions of this bilinear
Hirota equation, which are given in terms of elliptic hypergeometric
sum/integrals.
