A Twisted sl_2(C) Isomonodromic-Isospectral Correspondence

The aim of this article is to generalize the isomonodromic-isospectral correspondence for meromorphic connections of rank $2$ over $\mathbb{P}^1$ to the twisted case. More specifically, the construction of the isospectral approach is provided for the Painlev\'{e} I hierarchy, then two maps are constructed, one linking the sets of isomonodromic and isospectral Hamiltonians, another linking the set of apparent singularities to a set of isospectral coordinates.