Integrable differential systems of topological type and reconstruction by the topological recursion

Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x \Psi= L\Psi$and$\hbar \partial_t \Psi=R\Psi$ we prove that, under certain assumptions (genus $0$ spectral curve and additional conditions on $R$ and $L$), the system satisfies the “topological type property”. A consequence is that the formal $\mathrm{\hslash }$-WKB expansion of its determinantal correlators, satisfy the topological recursion. This applies in particular to all $(p,q)$ minimal models reductions of the KP hierarchy, or to the six Painlev'e systems.