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An r-matrix formalism is applied to the construction of the integrable lattice systems and their bi-Hamiltonian structure. Miura-like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.

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Geometric separability theory of Gel'fand-Zakharevich bi-Hamilto-nian systems on Riemannian manifolds is reviewed and developed. Particular attention is paid to the separability of systems generated by the so-called special conformal Killing tensors, i.e. Benenti systems. Then, infinitely many new classes of separable systems are constructed by appropriate deformations of Benenti class systems. All such systems can be lifted to the Gel'fand-Zakharevich bi-Hamiltonian form.