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We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at $${4m+12}$$ points for $${m \geq 1}$$ , which appear in pairs due to a symmetry condition. We parameterize this linear system in terms of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation. [ABSTRACT FROM AUTHOR]