  We study the moduli space of trace-free irreducible rank 2 holomorphic
connections over a complex projective 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 allow 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
Poincar\'e 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 vanGeemen-Previato. We explicitly describe the
isomonodromic foliation in the moduli space of vector bundles with
$\mathfrak{sl}_2({\mathbb C})$-connection over curves of genus 2 and prove the
transversality of the induced flow with the locus of unstable bundles.
