Nonlinear evolution equations and their stationary reductions

In recent years there have been many papers on stationary flows of integrable nonlinear evolution equations and their Hamiltonian properties. In particular there have been some results concerning the reversal of the roles of $x$ and $t$, resulting in PDEs which are Hamiltonian and give the usual stationary Poisson brackets in the reduced case. To date the results have been rather ad hoc and disparate. In this brief report we give a systematic construction of these $x-t$ reversed equations and their Hamiltonian properties, using their isospectral properties. We illustrate our approach with examples from the KdV hierarchy. Bibl. 5 titles.