The Painlevé equations are classified into eight types, and all of them are obtained from degenerations of the sixth Painlevé equation. This is the case of a two-dimensional phase space. When we consider the four-dimensional case, we have four systems that are sources, playing the same role that the sixth Painlevé equation does for the two-dimensional case. In this paper, we write down the Hamiltonian equations corresponding to all of the four systems that are obtained from the deformation theory of the Fuchsian equations. These are the well-known Garnier system with two independent variables, a Fuji-Suzuki system, a Sasano system, and the last one, which is new.

This is a continuation of Part 1 by one of the authors. We have shown that there are four 4-dimensional Painlevé-type equations derived from isomonodromic deformation of the Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlevé system. In Part 2, we degenerate these four source equations, and systematically obtain other 4-dimensional Painlevé-type equations, whose associated linear equations are of unramified type. There are 22 types of 4-dimensional Painlevé-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as the Noumi-Yamada systems are included in this list. They are expressed as Hamiltonian systems, and their Hamiltonians are simply written by using the Hamiltonians of the classical Painlevé equations.

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. We shall give an interpretation of the list of 4-dimensional Painlevé type equations given by Kawakami-Nakamura-Sakai from our classification of 4-dimensional moduli spaces of connections. In the study of the symmetries, a realization of the moduli spaces as quiver varieties is given and plays an essential role.

In this appendix, we provide a list of Hamiltonians for 4-dimensional Painlevé-type equations derived from the isomonodromic deformation of linear equations of ramified type (see Section 1.2 in Part 2).