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) (...

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, d...

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized ...