Classical Poisson structures andr-matrices from constrained flows

We construct the classical Poisson structures and r-matrices for some finite-dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows us to produce new kinds of classical (dynamical) Yang - Baxter structures. To illustrate this method we present the r-matrices associated with the constrained flows of the Kaup - Newell, KdV, WKI and TG hierarchies, all generated by a two-dimensional eigenvalue problem. Some of the r-matrices thus obtained depend only on the spectral parameters, but others depend also on the dynamical variables. For consistency they have to obey a classical Yang - Baxter-type equation, possibly with dynamical extra terms.