A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map $RH:ℳ\to ℛ$, where $ℳ$ is a moduli space of connections and $ℛ$, the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of $RH$ (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces $ℳ$. The induced Painlevé equations are computed explicitly....