Poisson Pencils, Lie Symmetries and Hamiltonian Reductions of a Coupled Nonlinear Wave System:Tangent KdV Geometry and Elliptic Moduli

We study the two-field nonlinear dispersive system \[ u_t=u_{xxx}+6uu_x,\qquad v_t=v_{xxx}+6(uv)_x, \] viewed simultaneously as a coupled wave equation, as the tangent covering of Korteweg--de Vries (KdV), and as a Hamiltonian flow on a tangent Poisson manifold. The main purpose is to make these v