  As a development of arXiv:1912.12897, we note that the ordinary Shiraishi
functions have an insufficient number of parameters to describe generic
eigenfunctions of double elliptic system (Dell). The lacking parameter can be
provided by substituting elliptic instead of the ordinary Gamma functions in
the coefficients of the series. These new functions (ELS-functions) are
conjectured to be functions governed by compactified DIM networks which can
simultaneously play the three roles: solutions to non-stationary Dell
equations, Dell conformal blocks with the degenerate field (surface operator)
insertion, and the corresponding instanton sums in $6d$ SUSY gauge theories
with adjoint matter. We describe the basics of the corresponding construction
and make further conjectures about the various limits and dualities which need
to be checked to make a precise statement about the Dell description by
double-periodic network models with DIM symmetry. We also demonstrate that the
ELS-functions provide symmetric polynomials, which are an elliptic
generalization of Macdonald ones, and compute the generation function of the
elliptic genera of the affine Laumon spaces. In the particular $U(1)$ case, we
find an explicit plethystic formula for the $6d$ partition function, which is a
non-trivial elliptic generalization of the $(q,t)$ Nekrasov-Okounkov formula
from $5d$.
