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In this paper a list of $R$-matrices on a certain coupled Lie algebra isobtained. With one of these $R$-matrices, we construct infinitely manybi-Hamiltonian structures for each of the two-component BKP and the Todalattice hierarchies. We also show that, when such two hierarchies are reducedto their subhierarchies, these bi-Hamiltonian structures are reducedcorrespondingly.

$R$-matrices and Hamiltonian Structures for Certain Lax Equations