  In this article we investigate stationary cKdV systems and prove that every
$N$-field stationary cKdV system can be written, after a careful
reparametrization of jet variables, as a classical separable St\"ackel system
on $N+1$ different ways. For each of these $N+1$ parametrizations we present an
explicit map between the jet variables and the separation variables of the
system. Finally, we show that each pair of St\"ackel 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.
