In this paper, we build the Hamiltonian system and the corresponding Lax
pairs associated to a twisted connection in gl2​(C)
admitting an irregular and ramified pole at infinity of arbitrary degree, hence
corresponding to the Painlev\'{e} 1 hierarchy. We provide explicit formulas
for these Lax pairs and Hamiltonians in terms of the irregular times and
standard 2g Darboux coordinates associated to the twisted connection.
Furthermore, we obtain a map that reduces the space of irregular times to only
g non-trivial isomonodromic deformations. In addition, we perform a
symplectic change of Darboux coordinates to obtain a set of symmetric Darboux
coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we
apply our general theory to the first cases of the hierarchy: the Airy case
(g=0), the Painlev\'{e} 1 case (g=1) and the next two elements of the
Painlev\'{e} 1 hierarchy.Comment: 44 pages + appendices. arXiv admin note: text overlap with
arXiv:2212.0483