  In this paper, we study and build the Hamiltonian system attached to any
$\mathfrak{gl}_2(\mathbb{C})$ meromorphic connection with 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 Painlev\'{e} equations. We finally
make the connection with the topological recursion and the quantization of
classical spectral curve from this perspective.
