Stationary coupled KdV systems and their Stäckel representations

In this article, we investigate stationary coupled Korteweg–de Vries (cKdV) systems and prove that every $N$-field stationary cKdV system can be written, after a careful reparameterization of jet variables, as a classical separable Stäckel system in $N+1$ different ways. For each of these $N+1$ parameterizations, we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of Stäckel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite-dimensional Miura map, which leads to an $(N+1)$-Hamiltonian formulation for the stationary cKdV system.