The differential equations governing isomonodromic deformations of systems of linear differential equations play a crucial role in integrable systems. Similarly, discrete isomonodromic deformations of systems of linear difference equations play a similar role in the theory of discrete integrable systems. These difference equations can be of h-difference, q-difference and elliptic difference type. The latter is interpreted as a linear system of difference equations whose entries are parameterized in terms of theta functions. We present a system of discrete isomonodromic deformations of a second order system of linear elliptic difference equations with 4m+12 singular points which we call an elliptic Garnier system. The lowest nontrivial case, when m=1, is identified with the elliptic Painleve equation, hence, our work provides a new an novel Lax pair for the elliptic Painleve equation.
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