In this note, we give some holomorphy conditions of Fuji-Suzuki coupled Painleve VI system. We also give two translation operators acting on the constant parameter $eta$. We note a confluence process from the Fuji-Suzuki system to the Noumi-Yamada system of type $A_5^{(1)}$.
In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables $q_1,q_2$, however
, has same five holomorphy conditions in the variables $p_1,p_2$.
In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painleve I equation. We will show that this system passes the Painleve $alpha$-test for all accessible sin
gular points $P_i (i=1,2,3)$. We note its holomorphy condition of the first Painleve system.
We remark on the Garnier system in two variables.
In this note, we review the notion of Painleve scheme of the sixth Painleve equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${Sigma_2}$. The key method is Painleve $alpha$-method fo
r each accessible singular point. Giving a Painleve scheme in the differential system satisfying certain conditions, we can recover the Painleve VI system with the polynomial Hamiltonian. We also consider the case of the Painleve V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular $(n+2)$-points in the Hirzebruch surface of degree $n$; ${Sigma_n}$. This equation has symmetry of symmetric group of degree $n+2$.
We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three variables. W
e give a relation between it and the second member of the second Painleve hierarchy. Moreover, we give some relations between an autonomous version of its polynomial Hamiltonian system in two variables and the mKdV hierarchies.
We present {it symmetric Hamiltonians} for the degenerate Garnier systems in two variables. For these symmetric Hamiltonians, we make the symmetry and holomorphy conditions, and we also make a generalization of these systems involving symmetry and ho
lomorphy conditions inductively. We also show the confluence process among each system by taking the coupling confluence process of the Painleve systems.
We find and study a two-parameter family of coupled Painleve II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables $t,s$. Setting $
s=0$, we can obtain an autonomous version of the coupled Painleve II systems. We also show its symmetry and holomorphy conditions.
We find a two-parameter family of coupled Painleve 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 Painleve systems in dimens
ion 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 $A_1^{(1)}$.
We study a one-parameter family of the fourth-order ordinary differential equations obtained by similarity reduction of the modifed Sawada-Kotera equation. We show that the birational transformations take this equation to the polynomial Hamiltonian s
ystem in dimension four. We make this polynomial Hamiltonian from the viewpoint of accessible singularity and local index. We also give its symmetry and holomorphy conditions. These properties are new. Moreover, we introduce a symmetric form in dimension five for this Hamiltonian system by taking the two invariant divisors as the dependent variables. Thanks to the symmetric form, we show that this system admits the affine Weyl group symmetry of type $A_2^{(2)}$ as the group of its B{a}cklund transformations.
