Studies on a Five-parameter Family of Four-dimensional Partial Differential Systems in Two Variables

We find and study a five-parameter family of four-dimensional partial differential systems in two variables from the viewpoints of its symmetry and holomorphy properties. 1. Main results In this paper, we present a 5-parameter family of four-dimensional partial differential systems in two variables explicitly given by (cf. [3, 18]) 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 = HV I(q1, p1, t;α1, α2, α3, α4) + α2p2{ −(t− 1)sq1 + t(s− 1)q2 + (t− s)q1q2 t(t− 1)(t− s) + (q1 − t)q2(q2 − 1) t(t− 1)(t− η) } + α5p1{ (t− s)q1(q1 − 1) + t(t− 1)(q1 − q2) t(t− 1)(t− s) + (q1 − t)((t− 1)q1 + (q1 − t)q2) t(t− 1)(t− η) } − p1p2{ (t− 1)(sq 1 + tq 2 2)− (t− s)q2(q 2 1 + t)− 2t(s− 1)q1q2 t(t− 1)(t− s) − (q1 − t) q2(q2 − 1) t(t− 1)(t− η) } + α2α5(2tq1 − q1 − tq2 + q1q2 − ηq1) t(t− 1)(t− η) , H2 = π(H1) (α0 + α1 + 2α2 + α3 + α4 + 2α5 = 1), (1) where the transformation π is explicitly given by π :(q1, p1, q2, p2, t, s;α0, α1, α2, α3, α4, α5) → (q2, p2, q1, p1, s, t;α0, α1, α5, α3, α4, α2). (2)