Coupled Painlev\'e II Systems in Dimension 4 and the systems of type A4(1){A_4}^{(1)}

We find and study coupled Painlev\'e II systems in dimension 4, which can be obtained by a degeneration from the systems of type ${A_4}^{(1)}$. We compare these systems with other types of coupled Painlev\'e II systems from the viewpoint of the local index. We also give the phase spaces for these systems.Comment: This paper has been withdrawn by the author. 14 pages, 2 figure

Double covering of the Painlev\'e I equation and its singular analysis

In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painlev\'e I equation. We will show that this system passes the Painlev\'e $\alpha$-test for all accessible singular points $P_i \ (i=1,2,3)$. We note its holomorphy condition of the first Painlev\'e system.Comment: 18 pages, 4 figure

Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication

The aim of this paper is to study relations between regular reductive PVs with one-dimensional scalar multiplication and the structure of graded Lie algebras. We will show that the regularity of such PVs is described by an $\mathfrak{sl}_2$-triplet of a graded Lie algebra

Remark on the Garnier system in two variables

We remark on the Garnier system in two variables.Comment: 4 page

Four-dimensional Painlev\'e systems of types D5(1)D_5^{(1)} and B4(1)B_4^{(1)}

We find and study a five-parameter family of four-dimensional coupled Painlev\'e V systems with affine Weyl group symmetry of type $D_5^{(1)}$. We then give an explicit description of a confluence from those systems to a four-parameter family of four-dimensional coupled Painlev\'e III systems with affine Weyl group symmetry of type $B_4^{(1)}$.Comment: 19 pages, 6 figure

Reduced contragredient Lie algebras and PC Lie algebras

The first aim of this paper is to show that any finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation can be embedded into some PC Lie algebra. The second aim is to find the structure of a PC Lie algebra

Coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of type D6(1)D_6^{(1)}, II

We give a reformulation of a six-parameter family of coupled Painlev\'e VI systems with affine Weyl group symmetry of type $D_6^{(1)}$ from the viewpoint of its symmetry and holomorphy properties.Comment: 14 pages, 5 figure

Coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of types A4(2)A_4^{(2)} and A1(1)A_1^{(1)}

We find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $A_4^{(2)}$. For a degenerate system of $A_4^{(2)}$ system, we also find a one-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $A_1^{(1)}$. We show that for each system, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new. Moreover, we find a one-parameter family of partial differential systems in three variables with $W(A_1^{(1)})$-symmetry. We show the relation between its polynomial Hamiltonian system and an autonomous version of the system of type $A_1^{(1)}$.Comment: 12 pages, 1 figur

Coupled Painlev\'e III systems with affine Weyl group symmetry of types B4(1)B_4^{(1)}, D4(1)D_4^{(1)} and D5(2)D_5^{(2)}

We find and study four kinds of a 4-parameter family of four-dimensional coupled Painlev\'e III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$. We also show that these systems are equivalent by an explicit birational and symplectic transformation, respectively.Comment: 13 pages, 5 figure

Coupled Painlev\'e systems with affine Weyl group symmetry of types D3(2)D_3^{(2)} and D5(2)D_5^{(2)}

In this paper, we find a two-parameter family of coupled Painlev\'e systems in dimension four with affine Weyl group symmetry of type $D_3^{(2)}$. We also find a four-parameter family of 2-coupled $D_3^{(2)}$-systems in dimension eight with affine Weyl group symmetry of type $D_5^{(2)}$. We show that for each system, we give its symmetry and holomorphy conditions, respectively. These symmetries, holomorphy conditions and invariant divisors are new.Comment: 8 page