A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation are revealed. A class of its rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed. The latter lead to large families of biorthogonal rational functions based upon the very-well-poised balanced hypergeometric series of three types: the standard hypergeometric series ₉ F ₈, basic series ₁₀ϕ₉ and its elliptic analogue ₁₀ E ₉. For an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly.