Hamiltonian representation of isomonodromic deformations of general rational connections on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>gl</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml

Abstract In this paper, we study and build the Hamiltonian system attached to any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="fraktur">gl</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> meromorphic connection with an arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs and Hamiltonian evolutions expressed in terms of irregular times and monodromies associated to the poles as well as g Darboux coordinates defined as the apparent singularities arising in the oper gauge. Moreover, we also provide a reduction of the isomonodromic deformations to a subset of g non-trivial isomonodromic deformations. This reduction is equivalent to a map reducing the set of irregular times to only g non-trivial isomonodromic times. We apply our construction to all cases where the associated spectral curve has genus 1 and recover the standard Painlevé equations. We finally make the connection with the topological recursion and the quantization of classical spectral curve from this perspective.