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"We study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus 2 case, connections as above are invariant under the hyperelliptic involution: they descend as rank 2 logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allows us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical (16, 6)-configuration of the Kummer surface. We also recover a Poincarape family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by van Geemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl2-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles"--

Vector bundles. --- Higgs bosons. --- Fibrés vectoriels. --- Higgs, Bosons de. --- Moduli theory. --- Differential equations, Parabolic. --- Vector bundles --- Moduli theory --- Differential equations, Parabolic --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Fiber spaces (Mathematics)