Explicit Hamiltonian representations of meromorphic connections and duality from different perspectives: a case study

In this article, we present an explicit study of $\hbar$-deformed meromorphic connections in $\mathfrak{gl}_3(\mathbb{C})$ with an unramified irregular pole at infinity of order $r_\infty=3$ and their spectral dual corresponding to the $\mathfrak{gl}_2(\mathbb{C})$ Painlevé IV Lax pair. Using the ap