On the Polynomial Hamiltonian Structure of the Garnier System, Ii

We present a new expression of the polynomial Hamiltonian of the Garnier system in two variables. 1. Main results The Garnier system in two variables is equivalent to the rational Hamiltonian system given by (see [18], cf. [3]) dq1 = ∂H1 ∂p1 dt + ∂H2 ∂p1 ds, dp1 = − ∂H1 ∂q1 dt− ∂H2 ∂q1 ds, dq2 = ∂H1 ∂p2 dt + ∂H2 ∂p2 ds, dp2 = − ∂H1 ∂q2 dt− ∂H2 ∂q2 ds, H1 = − q1(q1 − 1)(q1 − t)(q1 − s)(q2 − t) (q1 − q2)t(t− 1)(t− s) {p21 + κ q1(q1 − 1) − ( θ1 − 1 q1 − t + θ2 q1 − s + κ0 q1 + κ1 q1 − 1 ) p1} − q2(q2 − 1)(q2 − t)(q2 − s)(q1 − t) (q2 − q1)t(t− 1)(t− s) {p22 + κ q2(q2 − 1) − ( θ1 − 1 q2 − t + θ2 q2 − s + κ0 q2 + κ1 q2 − 1 )