  We derive the $2$d Zakharov-Mikhailov action from $4$d Chern-Simons theory.
This $2$d action is known to produce as equations of motion the flatness
condition of a large class of Lax connections of Zakharov-Shabat type, which
includes an ultralocal variant of the principal chiral model as a special case.
At the $2$d level, we determine for the first time the covariant Poisson
bracket $r$-matrix structure of the Zakharov-Shabat Lax connection, which is of
rational type. The flatness condition is then derived as a covariant Hamilton
equation. We obtain a remarkable formula for the covariant Hamiltonian in term
of the Lax connection which is the covariant analogue of the well-known formula
"$H=Tr L^2$".
