  We consider non-twisted meromorphic connections in
$\mathfrak{sl}_2(\mathbb{C})$ and the associated symplectic Hamiltonian
structure. In particular, we provide explicit expressions of the Lax pair in
the geometric gauge supplementing previous results where explicit formulas have
been obtained in the oper gauge. Expressing the geometric Lax matrices requires
the introduction of specific Darboux coordinates for which we provide the
explicit Hamiltonian evolutions. These expressions allow to build bridges
between the isomonodromic deformations and the isospectral ones. More
specifically, we propose an explicit change of Darboux coordinates to obtain
isospectral coordinates for which Hamiltonians match the spectral invariants.
This result solves the issue left opened in \cite{BertolaHarnadHurtubise2022}
in the case of $\mathfrak{sl}_2(\mathbb{C})$.
