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Description

We present a Lax pair for the sixth Painleve equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painleve equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painleve equation and the various degenerate versions of the third Painleve equation.