Commutation Relations and Discrete Garnier Systems

Understand

We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system.

• These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form.
• A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors.
• We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form.