The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

We introduce a way to represent pairs $(E,\nabla)$, where $E$ is a bundle on a Riemann surface and $\nabla$ is a logarithmic connection in $E$, based on a representation of the surface as the quotient of the exterior of the unit disc. In this representation, we write the local isomonodromic deformation conditions for the pairs $(E,\nabla)$. These conditions are written as a modified Schlesinger system on a Riemann sphere (reduced to the ordinary Schlesinger system in the typical case) supplemented by a certain system of linear equations.