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We reconcile the Hamiltonian formalism and the zero curvature representationin the approach to integrable boundary conditions for a classical integrablesystem in 1+1 space-time dimensions. We start from an ultralocal Poissonalgebra involving a Lax matrix and two (dynamical) boundary matrices.Sklyanin's formula for the double-row transfer matrix is used to deriveHamilton's equations of motion for both the Lax matrix {\bf and} the boundarymatrices in the form of zero curvature equations. A key ingredient of themethod is a boundary version of the Semenov-Tian-Shansky formula for thegenerating function of the time-part of a Lax pair. The procedure isillustrated on the finite Toda chain for which we derive Lax pairs of size$2\times 2$ for previously known Hamiltonians of type $BC_N$ and $D_N$corresponding to constant and dynamical boundary matrices respectively.

From Hamiltonian to zero curvature formulation for classical integrable boundary conditions