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

Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function